9.1.3How do I know if a series converges?

Convergence and Divergence

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Math Notes

Convergence and Divergence

Instead of saying this series “has a finite sum” or this series “has no finite sum,” mathematicians use the words “converges” and “diverges.” These words apply to sequences as well as series.

A sequence $u_{n}$ converges if $\lim\limits _ { n \rightarrow \infty } u _ { n }$ exists and is finite.

A sequence $u_{n}$ diverges if $\lim\limits _ { n \rightarrow \infty } u _ { n }$ does not exist or is infinite.

A series converges if it has a finite sum. This means that its sequence of partial sums has a finite limit.

The sum of a (convergent) infinite series is the limit of its sequence of partial sums.

By definition, $\displaystyle\sum _ { k = 1 } ^ { \infty } u _ { k } = \lim\limits _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } u _ { k }$ .

A series diverges if the limit of its sequence of partial sums does not exist.

9-32.

If a sequence converges, does its corresponding series converge? Consider the sequence $u _ { n } = \frac { 1 } { \operatorname { ln } ( n + 1 ) }$ .

Does the sequence converge? In other words, does the $n$ ^th term of this infinite sequence exist and is it finite? Use your calculator to help you make your decision.
Does the series $\frac { 1 } { \operatorname { ln } ( 2 ) } + \frac { 1 } { \operatorname { ln } ( 3 ) } + \frac { 1 } { \operatorname { ln } ( 4 ) } + \cdots$ converge? Again, use your calculator to help you make your decision.

9-33.

Consider the series $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \ldots =$ $\displaystyle\sum _ { k = 1 } ^ { \infty } ( \frac { 1 } { 3 } ) ^ { k }$ .

Calculate the first four partial sums and write them as a sequence.
Demonstrate that the series $\displaystyle\sum _ { k = 1 } ^ { \infty } ( \frac { 1 } { 3 } ) ^ { k }$ has partial sums given by $S _ { n } = \frac { 1 } { 2 } ( 1 - \frac { 1 } { 3 ^ { n } } )$ .
Explain why the series converges and determine its sum by evaluating the limit of the sequence of partial sums as $n → ∞$ .
The diagram at right is often used to illustrate that the series $\frac { 1 } { 3 } + \frac { 1 } { 9 } + \frac { 1 } { 27 } + \ldots =$ $\displaystyle\sum _ { k = 1 } ^ { \infty } ( \frac { 1 } { 3 } ) ^ { k }$ . With your team, write an explanation about how this diagram illustrates that the sum you computed is correct.

Your teacher will provide you with a model.

9-34.

Give an example of an infinite series that diverges. Determine the limit of the sequence of its partial sums, then explain how this limit confirms that your series diverges.

9-35.

WHEN DO INFINITE GEOMETRIC SERIES CONVERGE?

Examine the infinite geometric series for the different common ratios below. With your team, test each of these series for convergence. Conjecture precisely which common ratios permit a series to converge, and which force it to diverge.
1. $\displaystyle \sum _ { n = 1 } ^ { \infty } 3 \cdot 2 ^ { n }$
1. $\displaystyle\sum _ { n = 1 } ^ { \infty } 3 \cdot 1 ^ { n }$
1. $\displaystyle\sum _ { n = 1 } ^ { \infty } 3 \cdot ( \frac { 1 } { 2 } ) ^ { n }$
1. $\displaystyle\sum _ { n = 1 } ^ { \infty } 3 \cdot ( - 2 ) ^ { n }$
1. $\displaystyle\sum _ { n = 1 } ^ { \infty } 3 \cdot ( - 1 ) ^ { n }$
1. $\displaystyle\sum _ { n = 1 } ^ { \infty } 3 \cdot ( - \frac { 1 } { 2 } ) ^ { n }$
Consider the general infinite geometric series $\displaystyle\sum _ { k = 1 } ^ { \infty } a r ^ { k - 1 }$ with $n$ ^th partial sum $S _ { n } = \frac { a - a r ^ { n } } { 1 - r }$ . Use the limit of the sequence of partial sums of an infinite series to determine the values of $r$ for which the series converges. Use the absolute value symbol to express these values of $r$ concisely.

9-36.

Now consider a series that is not geometric: $\frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \frac { 1 } { 20 }$

What is the $5$ ^th term of the series?
Write the series using sigma notation.
Determine the first three partial sums. Then write an equation for the $n$ ^th partial sum $S_n$ in terms of $n$ .
Does the series converge? If so, what is its sum?

9-37.

There is another way to determine the sum of the series from problem 9-36.

Return to the sigma notation you used in part (b) of problem 9-36. Rewrite the argument so that it is a sum (or difference) of two partial fractions. Then, without simplifying or combining like terms, write out the first ten terms of the series. What do you notice?
The series is now a telescoping series because when written in expanded form, most of the terms are eliminated. Use this idea to calculate the sum. Why do you think the word telescoping is used?
Rewrite the argument of each of the following telescoping series as the difference of two partial fractions, then calculate the sum.
1. $\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( n + 3 ) ( n + 5 ) }$
1. $\displaystyle\sum _ { n = 2 } ^ { \infty } \frac { 1 } { 4 n ^ { 2 } - 1 }$

9-38.

During a $10$ -minute race, Boris’ velocity in meters per minute was $v(t) = 0.5t^2 - 0.05t^3$ . At what time(s) was Boris’ velocity was equal to his average velocity? Homework Help ✎

9-39.

Draw a slope field for $\frac { d y } { d x }= 2xy$ for $−2 ≤ x ≤ 2$ . Homework Help ✎

If $y(-2) = 2$ , use Euler’s Method to draw a solution curve for $y$ using $∆x = 1$ .
Use implicit integration to solve for $y$ if $y(-2) = 2$ . Use your graphing calculator to graph this actual solution and compare to your solution graph from part (a).

9-40.

No calculator! A point is moving along the graph of $y = \cos(x)$ so that $\frac { d x } { d t }= 3$ . Determine the rate at which $y$ is changing when $x =\frac { \pi } { 4 }$ . Homework Help ✎

9-41.

No calculator! Evaluate the following limits. Homework Help ✎

$\lim\limits _ { x \rightarrow 0 }\ln(\cos(x))$

$\lim\limits _ { x \rightarrow 0 } e ^ { ( 1 - \operatorname { cos } ( x ) ) / x }$

$\lim\limits _ { x \rightarrow 0 }\ln( \frac { \operatorname { sin } ( x ) } { x ^ { 2 } + x } )$

$\lim\limits _ { x \rightarrow \infty } \frac { \operatorname { ln } ( x ) } { x ^ { 2 } + x }$

Compute without a calculator

9-42.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int _ { 0 } ^ { 1 } \frac { 1 + x } { \sqrt { 1 - x ^ { 2 } } } d x$

$\int _ { - \infty } ^ { \infty } \frac { 1 } { x } d x$

$\int _ { 0 } ^ { 1 } \frac { 1 - x } { x } d x$

$\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 3 ] { x } } d x$

9-43.

Multiple Choice: The length of the curve $y =\frac { x ^ { 3 } } { 3 }+ x^2$ for $2 ≤ x ≤ 5$ is: Homework Help ✎

$\frac { 13 } { 2 }$

$13$

$\frac { 27 } { 2 }$

$60.089$

9-44.

Determine whether each of the following infinite series converges or diverges. Explain each answer briefly. If the series converges, calculate its exact sum. Homework Help ✎

$\displaystyle \sum _ { k = 1 } ^ { \infty } 1$

$\displaystyle\sum _ { j = 2 } ^ { \infty } 8 ( \frac { 3 } { 4 } ) ^ { j }$

$\displaystyle\sum _ { k = 0 } ^ { \infty } 20 ( 0.4 ) ^ { k }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ( n + 1 ) }$

9-45.

A variant of Ying’s method (manipulating an infinite sum so that the sum appears as a part of itself) can be used in other situations. For example, to evaluate the infinite “nested radical” below, let: Homework Help ✎

$S = \sqrt { 2 + \sqrt { 2 + \sqrt { 2 + \sqrt { 2 + \ldots } } } }$

Explain why $S =\sqrt { 2 + S }$ . Then solve for $S$ .

Calculus Third Edition

Convergence and Divergence