Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Themes All Themes Colonialism and Nigerian Politics Religion and Belief Family Freedom vs. Tyranny Silence and Speech Violence. 1 day ago Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Themes All Themes Colonialism and Nigerian Politics Religion and Belief Family Freedom vs. Tyranny Silence and Speech Violence. Purple Notes 4.2 Purple Notes is a very simple, powerful and easy to use notes application. The application uses iCloud for the data synchronisation between your iPhone, iPad and Mac. Go to results section 3; Go to results section 4; Go to results section 5; Forward to more results section; Same-day pickup orders are typically ready in about 3 hours, but orders placed outside of store hours may require longer processing time or be ready the next day. Store availability is not guaranteed, and inventory may fluctuate. Springtrap is the only real animatronic featured in Five Nights at Freddy's 3. Springtrap's true identity is William Afton, the main villainous antagonist of the series and plays an important part in the hidden story of Five Nights at Freddy's. The suit is an olive green with a forest green belly patch as well as the inside of the ears, the suit has several holes ranging from small to large in.
PART ONE – BREAKING GODS: PALM SUNDAY
Summary
Palm Sunday marks a change in the Achike household. Narrator Kambili, the 15-year old daughter of a devout Catholic, is terrified of the punishment her brother Jaja will incur for missing the day's mass. When the family arrives home from church, Papa demands an explanation from his son. Why did he not receive communion? Jaja says it is because the wafer gives him bad breath. Papa is shocked and reminds Jaja that not accepting the Host – the body of their Lord – is death. Jaja responds that he will die. Papa then throws his heavy leather-bound missal across the room, missing Jaja but breaking his wife's beloved figurines.
Kambili's Papa Eugene is a revered member of Enugu, Nigeria. A prominent and wealthy business leader, Eugene is praised by St. Agnes' white Father Benedict for using his power to spread the Gospel and speak the truth. However, inside his own home, he is a feared authoritarian and strict disciplinarian. Kambili notes the fading black eye of her Mama Beatrice. Kambili still takes pride in her father and his deeds, though he urges the family to stay humble.
Tensions rise in the Achike house throughout the day. Jaja helps his Mama clean up the jagged pieces of the figurines while Papa has his tea. Kambili is dismayed that her father does not offer her a 'love sip' of his tea. Papa drinks quietly as if Jaja had not just talked back to him. Kambili goes up to her room and daydreams before lunch. She stares out over the expansive yard lined with frangipani, bougainvillea trees and hibiscus bushes. Mama's red hibiscuses are the pride of their parish. Each Sunday, flowers are plucked by Mama's prayer group members. Even the government officials who Jaja say try to bribe Papa cannot resist the hibiscus.
The usual Sunday routines do not occur. Mama does not plait Kambili's hair in the kitchen and Jaja does not go up to his own room to read before lunch. Kambili comes downstairs when lunch is served by Sisi, the servant girl. Papa says grace over the meal, a ritual lasting more than twenty minutes. He addresses the Blessed Virgin as Our Lady, Shield of the People of Nigeria, a title he has invented. The meal proceeds in silence until Mama mentions that a new product has been delivered to the house that afternoon – bottles of cashew juice from one of Papa's factories.
Papa pours a glass of the yellow liquid for each member of the family. Kambili hopes that if she praises the juice, Papa will forget that he has not yet punished Jaja for his insubordination. Both Kambili and Mama offer kind words to Papa about the juice. Jaja says nothing. Papa stares at his son and again demands an explanation. Jaja says he has no words in his mouth. He then excuses himself before Papa can give the final prayer. Kambili swallows all of her cashew juice and has a severe coughing fit.
Kambili spends the rest of the night sick in her room. Both Papa and Mama come to check on her, but she is nauseated and deep in thought about her brother. Mama offers her some soup, but Kambili vomits. She asks about Jaja, who did not visit her after dinner. Mama tells her daughter that Jaja did not come down for supper either. Kambili then asks about Mama's figurines. Mama will not replace them.
Kambili lies in bed and realizes that Papa's missal did not just break Mama's figurines. Everything was tumbling down. Kambili thinks Jaja's defiance is like the purple hibiscus in her Aunty Ifeoma's garden. They represent a new kind of freedom, unlike the chants of freedom shouted at the Government Center. The purple hibiscus represents a freedom to do and to be.
Analysis
Kambili narrates the book in the first person, but in the past tense. The book has a unique structure that begins with the events of Palm Sunday, as described in the first chapter. The next twelve chapters chronicle the events that culminate in Jaja skipping communion on Palm Sunday. The following four chapters detail the immediate aftermath of Palm Sunday. The final chapter, which is the indicated as the present, is three years after the events of the rest of the novel. Kambili, now eighteen years old, is narrating what happens to her and her family when she is fifteen. Through her eyes, we see the destruction of her family as well as the crumbling political situation of Nigeria. Told from a child's perspective, the novel is not overtly political and the debates on corruption unfold through conversation and overhearing. Since Kambili is not directly involved in activism, readers can draw their own conclusions about the political landscape from the personal experience of a young Nigerian. Her understanding of her family's pro-democracy stance is enhanced by her experiences with her liberal aunt.
Kambili's journey is a coming of age story set against multiple tyrannies. The corruption of her local government plays out in the background as Kambili is removed from direct strife due to her family's wealth. Her father's strict Catholic rule of their house is the greater tyranny Kambili must cope with. She alludes to emotions and events that will play out in the rest of the novel in the opening line, 'Things started to fall apart at home when my brother, Jaja, did not go to communion…' We know there is trouble to come since the opening paragraph contains an eruption of violence. Though we do not see any abuse in this first chapter, Kambili's fear is palpable. Her concern for the well-being of her brother signifies not only the punishments they have received in the past, but also that Jaja's behavior is new. This is a coming of age story for Jaja as well.
Religion is at the forefront of the Achike family. Kambili's faith is strong as she has been raised to be a devout Catholic girl. However, religion in Nigeria and also for Kambili is more complicated than it appears. The white image of God was brought over by colonialist British missionaries. Conversion to Catholicism for many Nigerians means eradicating their roots and traditions. The Achikes do not participate in any 'heathen' or 'pagan' rituals and are therefore singled out as model Catholics. Kambili is led to believe that anything traditional is evil, so she is severed from her ancestry. Kambili grows aware of the hypocrisy of her father's position as religious leader. Though he is praised for his commitment to the truth as published in his newspaper, the Achikes are forbidden to tell the truth about the situation in their own home. Papa's punishments are attempts to make his children perfect in the eyes of both the community and God. He does not enjoy abusing his family, but he believes he must correct their behavior. Mama is less severe than Papa, often pointing out the more beautiful, natural world of God. Kambili takes solace in the natural world, especially in her mother's famous red hibiscuses. Mama's connection with nature and respect for the natural world represents another dimension of faith. Mama finds God in the natural world, not just in the rosary. Kambili's relationship with God is complex, consisting of the fear of hell instilled by Papa and the reverence for beauty instilled by Mama.
Their relationship with Papa is complex as well. Though it is clear that her father rules their household with an iron fist, a deep love for her Papa is evident. She swells with pride when Father Benedict praises Papa's deeds and charity. Kambili represents modern Africa, at a crossroads between colonial faith and traditional views. Her church does not allow any worship in Igbo, their native language. There is constant tension between the Igbo rituals and the rigid, Western mores of Catholicism. Jaja's heresy and insubordination is startling and Kambili becomes ill from the stress. Her coughing fit at dinner is a physical reaction to the change that has come over Jaja. As explored more fully in the next section, Kambili's repression manifests itself in a loss of words.
Jaja and Mama's actions are symbolic of the events that will unfold throughout the rest of the novel. When talking to her mother after supper, Kambili notes the recent scar on her face. Mama is a victim of Papa's abuse, but there is a sense that she will be putting a stop to the violence. As illustrated in the following section, Mama's figurines are a source of escapism from the tensions of home life. When she tells Kambili that she will not replace them, it is a signal that she is facing reality. Jaja's back-talk to his father signifies that he will no longer adhere to a faith he does not believe in simply because he is threatened by violence. Both Jaja and Mama are standing up to Papa.
Show Mobile NoticeShow All NotesHide All NotesSection 4-4 : Convergence/Divergence of Series
In the previous section we spent some time getting familiar with series and we briefly defined convergence and divergence. Before worrying about convergence and divergence of a series we wanted to make sure that we've started to get comfortable with the notation involved in series and some of the various manipulations of series that we will, on occasion, need to be able to do.
As noted in the previous section most of what we were doing there won't be done much in this chapter. So, it is now time to start talking about the convergence and divergence of a series as this will be a topic that we'll be dealing with to one extent or another in almost all of the remaining sections of this chapter.
So, let's recap just what an infinite series is and what it means for a series to be convergent or divergent. We'll start with a sequence (left{ {{a_n}} right}_{n = 1}^infty ) and again note that we're starting the sequence at (n = 1) only for the sake of convenience and it can, in fact, be anything.
Next, we define the partial sums of the series as,
[begin{align*}&{s_1} = {a_1} & {s_2} = {a_1} + {a_2} & {s_3} = {a_1} + {a_2} + {a_3} & {s_4} = {a_1} + {a_2} + {a_3} + {a_4} & hspace{0.25in}, vdots & {s_n} = {a_1} + {a_2} + {a_3} + {a_4} + cdots + {a_n} = sumlimits_{i = 1}^n {{a_i}} end{align*}]and these form a new sequence, (left{ {{s_n}} right}_{n = 1}^infty ).
An infinite series, or just series here since almost every series that we'll be looking at will be an infinite series, is then the limit of the partial sums. Or,
[sumlimits_{i = 1}^infty {{a_i}} = mathop {lim }limits_{n to infty } {s_n}]It is important to remember that (sumlimits_{i = 1}^infty {{a_i}} ) is really nothing more than a convenient notation for (mathop {lim }limits_{n to infty } sumlimits_{i = 1}^n {{a_i}} ) so we do not need to keep writing the limit down. We do, however, always need to remind ourselves that we really do have a limit there!
If the sequence of partial sums is a convergent sequence (i.e. its limit exists and is finite) then the series is also called convergent and in this case if (mathop {lim }limits_{n to infty } {s_n} = s) then, (sumlimits_{i = 1}^infty {{a_i}} = s). Likewise, if the sequence of partial sums is a divergent sequence (i.e. its limit doesn't exist or is plus or minus infinity) then the series is also called divergent.
Let's take a look at some series and see if we can determine if they are convergent or divergent and see if we can determine the value of any convergent series we find.
Example 1 Determine if the following series is convergent or divergent. If it converges determine its value. [sumlimits_{n = 1}^infty n ] Show SolutionTo determine if the series is convergent we first need to get our hands on a formula for the general term in the sequence of partial sums.
[{s_n} = sumlimits_{i = 1}^n i ]This is a known series and its value can be shown to be,
[{s_n} = sumlimits_{i = 1}^n i = frac{{nleft( {n + 1} right)}}{2}]Don't worry if you didn't know this formula (we'd be surprised if anyone knew it…) as you won't be required to know it in my course.
So, to determine if the series is convergent we will first need to see if the sequence of partial sums,
[left{ {frac{{nleft( {n + 1} right)}}{2}} right}_{n = 1}^infty ]is convergent or divergent. That's not terribly difficult in this case. The limit of the sequence terms is,
[mathop {lim }limits_{n to infty } frac{{nleft( {n + 1} right)}}{2} = infty ]Therefore, the sequence of partial sums diverges to (infty ) and so the series also diverges.
So, as we saw in this example we had to know a fairly obscure formula in order to determine the convergence of this series. In general finding a formula for the general term in the sequence of partial sums is a very difficult process. In fact after the next section we'll not be doing much with the partial sums of series due to the extreme difficulty faced in finding the general formula. This also means that we'll not be doing much work with the value of series since in order to get the value we'll also need to know the general formula for the partial sums.
We will continue with a few more examples however, since this is technically how we determine convergence and the value of a series. Also, the remaining examples we'll be looking at in this section will lead us to a very important fact about the convergence of series.
So, let's take a look at a couple more examples.
Example 2 Determine if the following series converges or diverges. If it converges determine its sum. [sumlimits_{n = 2}^infty {frac{1}{{{n^2} - 1}}} ] Show SolutionThis is actually one of the few series in which we are able to determine a formula for the general term in the sequence of partial fractions. However, in this section we are more interested in the general idea of convergence and divergence and so we'll put off discussing the process for finding the formula until the next section.
The general formula for the partial sums is,
[{s_n} = sumlimits_{i = 2}^n {frac{1}{{{i^2} - 1}}} = frac{3}{4} - frac{1}{{2n}} - frac{1}{{2left( {n + 1} right)}}]and in this case we have,
[mathop {lim }limits_{n to infty } {s_n} = mathop {lim }limits_{n to infty } left( {frac{3}{4} - frac{1}{{2n}} - frac{1}{{2left( {n + 1} right)}}} right) = frac{3}{4}]The sequence of partial sums converges and so the series converges also and its value is,
[sumlimits_{n = 2}^infty {frac{1}{{{n^2} - 1}}} = frac{3}{4}] Example 3 Determine if the following series converges or diverges. If it converges determine its sum. [sumlimits_{n = 0}^infty {{{left( { - 1} right)}^n}} ] Show SolutionIn this case we really don't need a general formula for the partial sums to determine the convergence of this series. Let's just write down the first few partial sums.
[begin{align*}&{s_0} = 1 & {s_1} = 1 - 1 = 0 & {s_2} = 1 - 1 + 1 = 1 & {s_3} = 1 - 1 + 1 - 1 = 0 & etc.end{align*}]So, it looks like the sequence of partial sums is,
[left{ {{s_n}} right}_{n = 0}^infty = left{ {1,0,1,0,1,0,1,0,1, ldots } right}]and this sequence diverges since (mathop {lim }limits_{n to infty } {s_n}) doesn't exist. Therefore, the series also diverges.
Example 4 Determine if the following series converges or diverges. If it converges determine its sum. [sumlimits_{n = 1}^infty {frac{1}{{{3^{n - 1}}}}} ] Show SolutionHere is the general formula for the partial sums for this series.
[{s_n} = sumlimits_{i = 1}^n {frac{1}{{{3^{i - 1}}}}} = frac{3}{2}left( {1 - frac{1}{{{3^n}}}} right)]Again, do not worry about knowing this formula. This is not something that you'll ever be asked to know in my class.
In this case the limit of the sequence of partial sums is,
[mathop {lim }limits_{n to infty } {s_n} = mathop {lim }limits_{n to infty } frac{3}{2}left( {1 - frac{1}{{{3^n}}}} right) = frac{3}{2}]The sequence of partial sums is convergent and so the series will also be convergent. The value of the series is,
[sumlimits_{n = 1}^infty {frac{1}{{{3^{n - 1}}}}} = frac{3}{2}]As we already noted, do not get excited about determining the general formula for the sequence of partial sums. There is only going to be one type of series where you will need to determine this formula and the process in that case isn't too bad. In fact, you already know how to do most of the work in the process as you'll see in the next section.
So, we've determined the convergence of four series now. Two of the series converged and two diverged. Let's go back and examine the series terms for each of these. For each of the series let's take the limit as (n) goes to infinity of the series terms (not the partial sums!!).
[begin{align*}& mathop {lim }limits_{n to infty } n = infty & hspace{0.75in}& {mbox{this series diverged}} & mathop {lim }limits_{n to infty } frac{1}{{{n^2} - 1}} = 0 & hspace{0.75in}& {mbox{this series converged}} & mathop {lim }limits_{n to infty } {left( { - 1} right)^n}{mbox{ doesn't exist}} & hspace{0.5in} & {mbox{this series diverged}} & mathop {lim }limits_{n to infty } frac{1}{{{3^{n - 1}}}} = 0 & hspace{0.75in} & {mbox{this series converged}}end{align*}]Notice that for the two series that converged the series term itself was zero in the limit. This will always be true for convergent series and leads to the following theorem.
Theorem
If (sum {{a_n}} ) converges then (mathop {lim }limits_{n to infty } {a_n} = 0).
Proof
First let's suppose that the series starts at (n = 1). If it doesn't then we can modify things as appropriate below. Then the partial sums are,
[{s_{n - 1}} = sumlimits_{i = 1}^{n - 1} {{a_i}} = {a_1} + {a_2} + {a_3} + {a_4} + cdots + {a_{n - 1}}hspace{0.25in}{s_n} = sumlimits_{i = 1}^n {{a_i}} = {a_1} + {a_2} + {a_3} + {a_4} + cdots + {a_{n - 1}} + {a_n}]
Next, we can use these two partial sums to write,
[{a_n} = {s_n} - {s_{n - 1}}]Now because we know that (sum {{a_n}} ) is convergent we also know that the sequence (left{ {{s_n}} right}_{n = 1}^infty ) is also convergent and that (mathop {lim }limits_{n to infty } {s_n} = s)for some finite value (s). However, since (n - 1 to infty ) as (n to infty ) we also have (mathop {lim }limits_{n to infty } {s_{n - 1}} = s).
We now have,
[mathop {lim }limits_{n to infty } {a_n} = mathop {lim }limits_{n to infty } left( {{s_n} - {s_{n - 1}}} right) = mathop {lim }limits_{n to infty } {s_n} - mathop {lim }limits_{n to infty } {s_{n - 1}} = s - s = 0]Be careful to not misuse this theorem! This theorem gives us a requirement for convergence but not a guarantee of convergence. In other words, the converse is NOT true. If (mathop {lim }limits_{n to infty } {a_n} = 0) the series may actually diverge! Consider the following two series.
[sumlimits_{n = 1}^infty {frac{1}{n}} hspace{0.75in}hspace{0.25in}sumlimits_{n = 1}^infty {frac{1}{{{n^2}}}} ]In both cases the series terms are zero in the limit as (n) goes to infinity, yet only the second series converges. The first series diverges. It will be a couple of sections before we can prove this, so at this point please believe this and know that you'll be able to prove the convergence of these two series in a couple of sections.
Again, as noted above, all this theorem does is give us a requirement for a series to converge. In order for a series to converge the series terms must go to zero in the limit. If the series terms do not go to zero in the limit then there is no way the series can converge since this would violate the theorem.
This leads us to the first of many tests for the convergence/divergence of a series that we'll be seeing in this chapter.
Divergence Test
If (mathop {lim }limits_{n to infty } {a_n} ne 0) then (sum {{a_n}} )will diverge.
Again, do NOT misuse this test. This test only says that a series is guaranteed to diverge if the series terms don't go to zero in the limit. If the series terms do happen to go to zero the series may or may not converge! Again, recall the following two series,
[begin{align*} & sumlimits_{n = 1}^infty {frac{1}{n}} & hspace{0.5in} & {mbox{diverges}} & sumlimits_{n = 1}^infty {frac{1}{{{n^2}}}} & hspace{0.5in} & {mbox{converges}}end{align*}]One of the more common mistakes that students make when they first get into series is to assume that if (mathop {lim }limits_{n to infty } {a_n} = 0) then (sum {{a_n}} ) will converge. There is just no way to guarantee this so be careful!
Let's take a quick look at an example of how this test can be used.
Example 5 Determine if the following series is convergent or divergent. [sumlimits_{n = 0}^infty {frac{{4{n^2} - {n^3}}}{{10 + 2{n^3}}}} ] Show SolutionWith almost every series we'll be looking at in this chapter the first thing that we should do is take a look at the series terms and see if they go to zero or not. If it's clear that the terms don't go to zero use the Divergence Test and be done with the problem.
That's what we'll do here.
[mathop {lim }limits_{n to infty } frac{{4{n^2} - {n^3}}}{{10 + 2{n^3}}} = - frac{1}{2} ne 0]The limit of the series terms isn't zero and so by the Divergence Test the series diverges.
The divergence test is the first test of many tests that we will be looking at over the course of the next several sections. You will need to keep track of all these tests, the conditions under which they can be used and their conclusions all in one place so you can quickly refer back to them as you need to.
Next we should briefly revisit arithmetic of series and convergence/divergence. As we saw in the previous section if (sum {{a_n}} ) and (sum {{b_n}} ) are both convergent series then so are (sum {c{a_n}} ) and (sumlimits_{n = k}^infty {left( {{a_n} pm {b_n}} right)} ). Furthermore, these series will have the following sums or values.
[sum {c{a_n}} = csum {{a_n}} hspace{0.75in}sumlimits_{n = k}^infty {left( {{a_n} pm {b_n}} right)} = sumlimits_{n = k}^infty {{a_n}} pm sumlimits_{n = k}^infty {{b_n}} ]We'll see an example of this in the next section after we get a few more examples under our belt. At this point just remember that a sum of convergent series is convergent and multiplying a convergent series by a number will not change its convergence.
We need to be a little careful with these facts when it comes to divergent series. In the first case if (sum {{a_n}} ) is divergent then (sum {c{a_n}} ) will also be divergent (provided (c) isn't zero of course) since multiplying a series that is infinite in value or doesn't have a value by a finite value (i.e. c) won't change the fact that the series has an infinite or no value. However, it is possible to have both (sum {{a_n}} ) and (sum {{b_n}} ) be divergent series and yet have (sumlimits_{n = k}^infty {left( {{a_n} pm {b_n}} right)} ) be a convergent series.
Now, since the main topic of this section is the convergence of a series we should mention a stronger type of convergence. A series (sum {{a_n}} ) is said to converge absolutely if (sum {left| {{a_n}} right|} ) also converges. Absolute convergence is stronger than convergence in the sense that a series that is absolutely convergent will also be convergent, but a series that is convergent may or may not be absolutely convergent.
In fact if (sum {{a_n}} )converges and (sum {left| {{a_n}} right|} ) diverges the series (sum {{a_n}} )is called conditionally convergent.
What Is 4 Minus Negative 3
At this point we don't really have the tools at hand to properly investigate this topic in detail nor do we have the tools in hand to determine if a series is absolutely convergent or not. So we'll not say anything more about this subject for a while. When we finally have the tools in hand to discuss this topic in more detail we will revisit it. Until then don't worry about it. The idea is mentioned here only because we were already discussing convergence in this section and it ties into the last topic that we want to discuss in this section.
In the previous section after we'd introduced the idea of an infinite series we commented on the fact that we shouldn't think of an infinite series as an infinite sum despite the fact that the notation we use for infinite series seems to imply that it is an infinite sum. It's now time to briefly discuss this.
First, we need to introduce the idea of a rearrangement. A rearrangement of a series is exactly what it might sound like, it is the same series with the terms rearranged into a different order.
For example, consider the following infinite series.
[sumlimits_{n = 1}^infty {{a_n}} = {a_1} + {a_2} + {a_3} + {a_4} + {a_5} + {a_6} + {a_7} + cdots ]
A rearrangement of this series is,
[sumlimits_{n = 1}^infty {{a_n}} = {a_2} + {a_1} + {a_3} + {a_{14}} + {a_5} + {a_9} + {a_4} + cdots ]The issue we need to discuss here is that for some series each of these arrangements of terms can have different values despite the fact that they are using exactly the same terms.
Here is an example of this. It can be shown that,
[begin{equation}sumlimits_{n = 1}^infty {frac{{{{left( { - 1} right)}^{n + 1}}}}{n}} = 1 - frac{1}{2} + frac{1}{3} - frac{1}{4} + frac{1}{5} - frac{1}{6} + frac{1}{7} - frac{1}{8} + cdots = ln 2label{eq:eq1}end{equation}]Since this series converges we know that if we multiply it by a constant (c) its value will also be multiplied by (c). So, let's multiply this by (frac{1}{2}) to get,
[begin{equation}frac{1}{2} - frac{1}{4} + frac{1}{6} - frac{1}{8} + frac{1}{{10}} - frac{1}{{12}} + frac{1}{{14}} - frac{1}{{16}} + cdots = frac{1}{2}ln 2 label{eq:eq2}end{equation}]Now, let's add in a zero between each term as follows.
[begin{equation}0 + frac{1}{2} + 0 - frac{1}{4} + 0 + frac{1}{6} + 0 - frac{1}{8} + 0 + frac{1}{{10}} + 0 - frac{1}{{12}} + 0 + cdots = frac{1}{2}ln 2label{eq:eq3}end{equation}]Note that this won't change the value of the series because the partial sums for this series will be the partial sums for the (eqref{eq:eq2}) except that each term will be repeated. Repeating terms in a series will not affect its limit however and so both (eqref{eq:eq2}) and (eqref{eq:eq3}) will be the same.
We know that if two series converge we can add them by adding term by term and so add (eqref{eq:eq1}) and (eqref{eq:eq3}) to get,
[begin{equation}1 + frac{1}{3} - frac{1}{2} + frac{1}{5} + frac{1}{7} - frac{1}{4} + cdots = frac{3}{2}ln 2 label{eq:eq4}end{equation}]
A rearrangement of this series is,
[sumlimits_{n = 1}^infty {{a_n}} = {a_2} + {a_1} + {a_3} + {a_{14}} + {a_5} + {a_9} + {a_4} + cdots ]The issue we need to discuss here is that for some series each of these arrangements of terms can have different values despite the fact that they are using exactly the same terms.
Here is an example of this. It can be shown that,
[begin{equation}sumlimits_{n = 1}^infty {frac{{{{left( { - 1} right)}^{n + 1}}}}{n}} = 1 - frac{1}{2} + frac{1}{3} - frac{1}{4} + frac{1}{5} - frac{1}{6} + frac{1}{7} - frac{1}{8} + cdots = ln 2label{eq:eq1}end{equation}]Since this series converges we know that if we multiply it by a constant (c) its value will also be multiplied by (c). So, let's multiply this by (frac{1}{2}) to get,
[begin{equation}frac{1}{2} - frac{1}{4} + frac{1}{6} - frac{1}{8} + frac{1}{{10}} - frac{1}{{12}} + frac{1}{{14}} - frac{1}{{16}} + cdots = frac{1}{2}ln 2 label{eq:eq2}end{equation}]Now, let's add in a zero between each term as follows.
[begin{equation}0 + frac{1}{2} + 0 - frac{1}{4} + 0 + frac{1}{6} + 0 - frac{1}{8} + 0 + frac{1}{{10}} + 0 - frac{1}{{12}} + 0 + cdots = frac{1}{2}ln 2label{eq:eq3}end{equation}]Note that this won't change the value of the series because the partial sums for this series will be the partial sums for the (eqref{eq:eq2}) except that each term will be repeated. Repeating terms in a series will not affect its limit however and so both (eqref{eq:eq2}) and (eqref{eq:eq3}) will be the same.
We know that if two series converge we can add them by adding term by term and so add (eqref{eq:eq1}) and (eqref{eq:eq3}) to get,
[begin{equation}1 + frac{1}{3} - frac{1}{2} + frac{1}{5} + frac{1}{7} - frac{1}{4} + cdots = frac{3}{2}ln 2 label{eq:eq4}end{equation}]Now, notice that the terms of (eqref{eq:eq4}) are simply the terms of (eqref{eq:eq1}) rearranged so that each negative term comes after two positive terms. The values however are definitely different despite the fact that the terms are the same.
Note as well that this is not one of those 'tricks' that you see occasionally where you get a contradictory result because of a hard to spot math/logic error. This is a very real result and we've not made any logic mistakes/errors.
Here is a nice set of facts that govern this idea of when a rearrangement will lead to a different value of a series.
Facts
Given the series(sum {{a_n}} ),
- If (displaystyle sum {{a_n}} ) is absolutely convergent and its value is (s) then any rearrangement of (displaystyle sum {{a_n}} ) will also have a value of (s).
- If (displaystyle sum {{a_n}} ) is conditionally convergent and (r) is any real number then there is a rearrangement of (displaystyle sum {{a_n}} ) whose value will be (r).
Purple Notes 4 3 X 4
Again, we do not have the tools in hand yet to determine if a series is absolutely convergent and so don't worry about this at this point. This is here just to make sure that you understand that we have to be very careful in thinking of an infinite series as an infinite sum. There are times when we can (i.e. the series is absolutely convergent) and there are times when we can't (i.e. the series is conditionally convergent).
As a final note, the fact above tells us that the series,
Purple Notes 4 3 0
[sumlimits_{n = 1}^infty {frac{{{{left( { - 1} right)}^{n + 1}}}}{n}} ]Purple Notes 4 3
must be conditionally convergent since two rearrangements gave two separate values of this series. Eventually it will be very simple to show that this series is conditionally convergent.
