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Chapter 8: Infinite Sequences and Series
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Section 8.2: Series

Essentials


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Notations such as $\sum _{n\={n}_{0}}^{\infty}{a}_{n}$, $\sum _{n}^{\infty}{a}_{n}$, or even $\sum ^{\infty}{a}_{n}$ and $\mathrm{\Σ}{a}_{n}$ are used to denote an infinite series, which is simply the sum of all the terms in an infinite sequence. (The simpler notations often appear in printed texts to save costs.)
As Zeno's paradox about the tortoise and the hare (actually, it was Achilles) shows, an infinite sequence of operations cannot logically be completed in a finite time, yet in reality it often is. So, by what rules is a meaning attached to a symbol that indicates the impossible task of executing an infinite number of additions?

Definitions


The key to giving meaning to the symbol $\stackrel{\infty}{\mathrm{\Σ}}{a}_{n}$ is the idea of the partial sum, as per Definition 8.2.1, below.
Definition 8.2.1: Partial Sums of an Infinite Series

The $N$th partial sum of the infinite series $\stackrel{\infty}{\mathrm{\Sigma}}{a}_{n}$ is ${S}_{N}\=\stackrel{N}{\mathrm{Sigma;}}{a}_{n}$, that is, the sum of the terms up through and including ${a}_{N}$. (Some texts will define ${S}_{N}$ as the sum of the first $N$ terms.)



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The meaning given to the symbol $\stackrel{\infty}{\mathrm{\Σ}}{a}_{n}$ is then $\underset{N\to \infty}{lim}{S}_{N}$, that is, the limit of the sequence of partial sums, $\left\{{S}_{N}\right\}$.
If this limit exists, then that number is the "sum" of the infinite series, and the series is said to converge to that number. If the limit of the sequence of partial sums does not exist, then the series is said to diverge. (An infinite series can diverge if the limit of the sequence of partial sums is infinite, or if it fails to exist because of oscillation.)
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Definitions 8.2.2 and 8.2.3 make these notions precise.
Definition 8.2.2: Convergence of an Infinite Series

•

The infinite series $\stackrel{\infty}{\mathrm{\Sigma}}{a}_{n}$ converges to $L$ if the limit of the sequence of partial sums ${S}_{N}\=\stackrel{N}{\mathrm{\Sigma}}{a}_{n}$converges to $L$.




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Definition 8.2.3: Divergence of an Infinite Series

•

The infinite series $\stackrel{\infty}{\mathrm{\Sigma}}{a}_{n}$ diverges if the limit of the sequence of partial sums ${S}_{N}\=\stackrel{N}{\mathrm{\Sigma}}{a}_{n}$ diverges, either because the limit is infinite, or because it oscillates.




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The astute reader will realize that Definitions 8.2.2 and 8.2.3 are reminiscent of the contents of Table 4.5.1 that detail similar calculations for improper integrals. There, the improper integral is defined as the limit of a proper integral, with the endpoint of integration approaching either infinity or a singularity on the real line.
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Definitions 8.2.4 and 8.2.5 refine the definition of convergence by distinguishing between absolute and conditional convergence.
Definition 8.2.4: Absolute Convergence

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If the series $\mathrm{\Σ}\left{a}_{n}\right$ converges, then the series $\mathrm{\Σ}{a}_{n}$ is said to be absolutely convergent.

•

If all the ${a}_{n}$ in a convergent series are nonnegative, then the series necessarily converges absolutely.




Definition 8.2.5: Conditional Convergence

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If $\mathrm{\Σ}{a}_{n}$ is a convergent series containing an infinite number of negative terms, but the series $\mathrm{\Σ}\left{a}_{n}\right$ diverges, then the convergence is said to be conditional.




Consequently, there is a certain ambiguity in declaring that an infinite series converges. Is it a series of nonnegative terms, in which case its convergence is necessarily absolute? Or if it contains an infinite number of negative terms, does it converge absolutely, or does it converge just conditionally? Hence, when discussing the convergence of an infinite series, this Study Guide will always modify the word "converge" with either conditionally, or absolutely, unless the context makes it perfectly clear that no such modification is needed to eliminate ambiguity.
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Theorems


Table 8.2.1 lists five theorems that summarize additional key points about the behavior of infinite series.
Theorem

Intuitive Statement

Formal Statement

8.2.1

•

An infinite series that converges absolutely, must necessarily converge conditionally.


If $\mathrm{\Σ}\left{a}_{n}\right$ converges, then $\mathrm{\Σ}{a}_{n}$ converges.

8.2.2

•

The general term of a convergent series must necessarily tend to zero.


If $\mathrm{\Σ}{a}_{n}$ converges, then ${a}_{n}\to 0$, but not conversely.

8.2.3

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Addition, subtraction and scalar multiplication for convergent sequences is wellbehaved.


If $\mathrm{\Σ}{a}_{n}equals;A$ and $\mathrm{\Σ}{b}_{n}equals;B$, then
1.

$\mathrm{\Σ}{a}_{n}\pm \mathrm{Sigma;}{b}_{n}equals;\mathrm{Sigma;}\left({a}_{n}\pm {b}_{n}\right)equals;Aplus;B$;

2.

$c{\mathrm{Sigma;a}}_{n}equals;\mathrm{Sigma;}c{a}_{n}equals;cA$, for any real number $c$.


8.2.4

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The Cauchy product of two convergent series may or may not converge. If the product does converge, it converges to the "right" value.

•

If one of the two series converges absolutely, then the product converges to the "right" value.

•

If both factors converge absolutely, then the product converges absolutely.


If $\underset{n\=0}{\mathrm{\Σ}}{a}_{n}equals;A$, $\underset{n\=0}{\mathrm{\Σ}}{b}_{n}equals;B$, and ${c}_{n}\=$$\sum _{k\=0}^{n}{a}_{k}{b}_{nk}$ , then
1.

$\mathrm{\Σ}{c}_{n}equals;A\cdot B$ if one of $\mathrm{\Σ}{a}_{n}$ or $\mathrm{\Σ}{b}_{n}$ converges absolutely.

2.

$\mathrm{\Sigma}{c}_{n}$ converges absolutely to $A\cdot B$ if both $\mathrm{\Σ}{a}_{n}$ and $\mathrm{\Σ}{b}_{n}$ converge absolutely.


8.2.5

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Any rearrangement or regrouping of the terms of an absolutely convergent series does not change the value of the sum.

•

The terms of a conditionally convergent series can be rearranged so the new sum is any desired real number.


Table 8.2.1 Relevant theorems for infinite series



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•

Theorem 8.2.1 is intuitively appealing because it simply says that if a sum of positive numbers converges, then making some of those numbers negative will at worst make the sum smaller.

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Theorem 8.2.2 is again somewhat intuitive in that an infinite sum that converges cannot have larger and larger terms in its "tail end." These tailend terms have to be getting smaller and smaller if the sum is to be finite. Now, the falsity of the converse, that if the $n$th term goes to zero then the series converges, is not intuitive. It takes a counterexample to show this. A standard counterexample is the socalled harmonic series, $\mathrm{\Σ}\left(1sol;n\right)$, which is shown to diverge by one of the devices developed in Chapter 8.3.

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Theorem 8.2.3 is a welcomed relief because it says that simple arithmetic with convergent series "works." Thus, addition, subtraction, and scalar multiplication of even conditionally convergent series produce new series that are convergent to the "right" values.

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•

Theorem 8.2.4 is a significant result because it both defines a method for forming a product between two infinite series, and because it also clarifies the conditions under which such a product of series results in a series that converges to the product of the values of the factors. The Cauchy product itself will be demonstrated at length in the Examples.

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Theorem 8.2.5 makes two statements, one about absolutely convergent series, and one about conditionally convergent series. The statement about absolutely convergent series shouldn't be surprising  such series are so well behaved that almost everything good about them is true. What's remarkable is the statement about conditionally convergent series, which can be made to converge to any real number by a suitable rearrangement of terms.



Some Types of Series


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Table 8.2.2 lists some examples and some types of series and their properties.
Series

Form

Properties

Geometric

$\sum _{n\=0}^{\infty}a{r}^{n}equals;\frac{a}{1r}$

Absolute convergence for $\leftr\right<1$${}$
${S}_{N}\=a\frac{1{r}^{Nplus;1}}{1r}$

$p$Series

$\sum _{n\=1}^{\infty}\frac{1}{{n}^{p}}$

Absolute convergence for $p\>1$
Diverges for $p\le 1$

Alternating

$\sum _{n\=0}^{\infty}{\left(1\right)}^{n}{a}_{n}$, with ${a}_{n}\>0$${}$

•

Converges if $\left\{{a}_{n}\right\}$ is decreasing with limit zero. (Leibniz)

•

If $S$ is the sum of the (convergent) series, and ${S}_{N}$ is the partial sum up through ${a}_{n}$, then $\leftS{S}_{N}\right\le {a}_{n\+1}$.


Harmonic

$\sum _{n\=1}^{\infty}\frac{1}{n}$

Diverges, even though ${a}_{n}\=\frac{1}{n}\to 0$

Alternating Harmonic

$\sum _{n\=1}^{\infty}\frac{{\left(1\right)}^{n\+1}}{n}$

Converges (conditionally) to $\mathrm{ln}\left(2\right)$

Telescoping

$\sum _{n\=0}^{\infty}\left({a}_{n}{a}_{n\+1}\right)$

${S}_{N}\={a}_{0}{a}_{N\+1}$
Converges (conditionally) if ${a}_{n}\to 0$

Table 8.2.2 Examples and types of series



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Examples


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Example 8.2.1

Sum the series$\sum _{n\=0}^{\infty}1\/{3}^{n}$ and show that the sum is the limit of the sequence of partial sums.

Example 8.2.2

Use Maple to sum the convergent $p$series $\sum _{n\=1}^{\infty}1\/{n}^{2}$ and show that the sum is the limit of the sequence of partial sums.

Example 8.2.3

a)

Use Maple to sum the alternating series $\sum _{n\=1}^{\infty}{\left(1\right)}^{n\+1}\/{n}^{2}$ and show that the sum is the limit of the sequence of partial sums.

b)

Test the claim that a partial sum is closer to the sum than the magnitude of the first neglected term.


Example 8.2.4

Sum the series $\sum _{n\=1}^{\infty}\frac{1}{n\left(nplus;1\right)}$ and show that the sum is the limit of the sequence of partial sums.

Example 8.2.5

Use Maple to sum the series $\sum _{n\=3}^{\infty}\frac{4}{{n}^{2}4}$ and show that the sum is the limit of the sequence of partial sums.

Example 8.2.6

Test the series $\sum _{n\=1}^{\infty}\mathrm{arctan}\left(n\right)$ for convergence.

Example 8.2.7

Test the series $\sum _{n\=1}^{\infty}\phantom{\rule[0.0ex]{5.0px}{0.0ex}}\mathrm{ln}\left(\frac{n}{5n\+2}\right)$ for convergence.

Example 8.2.8

Obtain the sum of the series $\sum _{n\=1}^{\infty}\phantom{\rule[0.0ex]{5.0px}{0.0ex}}\left(\mathrm{sin}\left(\frac{1}{n}\right)\mathrm{sin}\left(\frac{1}{n\+1}\right)\right)$ and show that the sum is the limit of the sequence of partial sums.

Example 8.2.9

Write the repeating decimal 3.$\stackrel{\&conjugate0;}{45}$ as the ratio of two integers.

Example 8.2.10

Write the repeating decimal 7.4$\stackrel{\&conjugate0;}{35}$ as the ratio of two integers.

Example 8.2.11

Use Maple to sum the series $\sum _{n\=1}^{\infty}\frac{1}{n\left(nplus;2\right)}$ and show that the sum is the limit of the sequence of partial sums.

Example 8.2.12

Use Maple to sum the series $\sum _{n\=1}^{\infty}\frac{1}{9{n}^{2}1}$ and show that the sum is the limit of the sequence of partial sums.
Note that although $\frac{1}{9{n}^{2}1}equals;\frac{1}{2}\left(\frac{1}{3n1}\frac{1}{3nplus;1}\right)$ (partial fractions), this is not a telescoping series.

Example 8.2.13

The Cauchy product of $\sum _{n\=0}^{\infty}{a}_{n}$ and $\sum _{n\=0}^{\infty}{b}_{n}$ is the series $\sum _{n\=0}^{\infty}{c}_{n}$, where ${c}_{n}\=\sum _{k\=0}^{n}{a}_{k}\cdot {b}_{nk}$.
What happens to ${c}_{n}$ when the index in both the series being multiplied starts not at $n\=0$, but $n\=1$?${}$

Example 8.2.14

Obtain the Cauchy product of $\sum _{n\=1}^{\infty}\frac{1}{{n}^{2}}$ with itself. Is the value of the product the square of the value of the given series?

Example 8.2.15

Obtain the Cauchy product of the absolutely convergent series $\sum _{n\=1}^{\infty}\frac{1}{{n}^{2}}$ and the conditionally convergent series $\sum _{n\=1}^{\infty}\frac{{\left(1\right)}^{n\+1}}{n}$. Is the product the product of the sums of the two given series?

Example 8.2.16

a)

Show that Leibniz' theorem on the convergence of alternating series applies to the alternating harmonic series. (See Table 8.2.2.)

b)

Use Maple to show that the sequence of partial sums converges to $\mathrm{ln}\left(2\right)$.

c)

Test the claim that a partial sum is closer to the sum than the magnitude of the first neglected term.


Example 8.2.17

a)

Show that Leibniz' theorem on the convergence of alternating series applies to the series $\sum _{n\=1}^{\infty}\phantom{\rule[0.0ex]{5.0px}{0.0ex}}\frac{{\left(1\right)}^{n\+1}}{\sqrt{n}}$. (See Table 8.2.2.)

b)

Obtain the first few, but graph the first 50, partial sums.

c)

If ${S}_{k}$ is the partial sum of the first $k$ terms, what value of $k$ will guarantee that the error in ${S}_{k}$ is no worse than ${10}^{3}$?





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