10.4.2Is absolute convergence absolutely necessary?

Regrouping and Rearranging Series

10-161.

Determine if each of the following series is divergent, absolutely convergent, or conditionally convergent.

$S = \displaystyle\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k + 1 } \frac { \operatorname { sin } ( \pi k ) } { 2 k }$

$S = \displaystyle\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k + 1 } \frac { \operatorname { ln } ( k + 1 ) } { k }$

10-162.

We have seen earlier that the graph of a power series often matches another function on a small interval. Compare the graph of the function $p ( x ) = - ( \frac { x ^ { 1 } } { 1 } + \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 3 } + \frac { x ^ { 4 } } { 4 } )$ to the graph of the function $f(x) = \ln(1 - x)$ .

For what interval do the two graphs appear to match each other well?
Extend the pattern you see in equation for $p(x)$ so that it is a tenth-degree polynomial and then compare the graphs of the new $y = p(x)$ and $y = f(x)$ . Does this change your answer for part (a)?
Write the new tenth-degree polynomial for $p(x)$ using sigma notation.
Compare $p_{10}(–1)$ and $f(–1)$ .
Alter the expression from part (c) to make $p(x)$ an even better approximation of $f(x) = \ln(1 − x)$ . Write this new series in sigma notation.

10-163.

In Lesson 10.4.1, you found that the alternating harmonic series $\displaystyle\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k + 1 } \frac { 1 } { k }$ is conditionally convergent. The Riemann Series Theorem states that a conditionally convergent series can be rearranged to converge to any number. How does this happen?

In the alternating harmonic series both the positive terms and negative terms of the series diverge, but the divergence of the positive terms is about the same as the divergence of the negative terms. Use your calculator to verify the following equality: $\displaystyle\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k + 1 } \frac { 1 } { k } =$ $\frac { 1 } { 1 } - \frac { 1 } { 2 } + \frac { 1 } { 3 } -$ $\frac{1}{4}+\ldots=\ln(2)$
A rearrangement of the terms is: $\frac{1}{1}+\frac{1}{3}-\frac{1}{2}+$ $\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+$ $\frac{1}{9}+\frac{1}{11}-\frac{1}{6}+$ $\frac{1}{13}+\frac{1}{15}+\ldots=$ $\ln(2\sqrt{2})$
Now the rearranged series gives the sum of the positive terms a little bit of a head start, so the outcome is a little larger. However, when a series is absolutely convergent, it is guaranteed to converge to one and only one value. Can you arrange the series in another way to get a different sum?

10-164.

For each of the following series, decide if it can be rearranged so that it converges to multiple values. Explain how you know this and indicate any convergence tests you use.

$\displaystyle \sum _ { k = 2 } ^ { \infty } ( - 1 ) ^ { k + 1 } \frac { k ^ { 2 } } { k ^ { 4 } + 2 k ^ { 3 } }$

$\displaystyle\sum _ { k = 0 } ^ { \infty } ( - 1 ) \frac { k ! } { 2 ^ { 2 k } }$

$\displaystyle\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { 2 ^ { k } }$

10-165.

The alternating harmonic series in problem 10-163, $\displaystyle \sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k + 1 } \frac { 1 } { k }$ , and the sigma notation you wrote to approximate $f(x) = \ln(1 − x)$ in part (e) of problem 10-162, $\displaystyle\sum _ { k = 1 } ^ { \infty } - \frac { x ^ { k } } { k }$ , should look similar.

Explain any differences you see in the two notations.
What value of $x$ will make the two series equivalent?

10-166.

Can the following series converge to multiple values? Explain how you know. Homework Help ✎

$5 - 3 + \frac { 9 } { 5 } - \frac { 27 } { 25 } +\ldots$

10-167.

When scientists talk about populations they often refer to the carrying capacity of species in a particular environment. Carrying capacity is the largest population that an environment can sustain forever.

Suppose the carrying capacity of seals for a particular group of islands is $2900$ and that there are currently $1800$ seals that inhabit the island. The rate of change of the number of seals is jointly proportional to the number of seals and the difference between the number of seals and the carrying capacity. Let $S$ represent the number of seals on the island after $t$ years. Homework Help ✎

Write a differential equation that models the rate of change in the number of seals.
Write a general solution for the differential equation.
Recall that $S = 1800$ when $t = 0$ because that is the current population. Suppose that after ten years, 2500 seals inhabit the island. Write a formula for $S$ in terms of $t$ .

10-168.

Determine the radius and interval of convergence for each of the following power series. Homework Help ✎

$\displaystyle \sum _ { k = 1 } ^ { \infty } \frac { ( - 4 ) ^ { k } ( x - 3 ) ^ { k } } { k }$

$\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { 2 ^ { k } ( x + 4 ) ^ { k } } { ( k + 1 ) ! }$

$1 + 3 x + \frac { 9 x ^ { 2 } } { 2 } + \frac { 27 x ^ { 3 } } { 3 } + \frac { 81 x ^ { 4 } } { 4 } +\ldots$

$\displaystyle\sum _ { k = 2 } ^ { \infty } \frac { x ^ { k } } { 3 ^ { k \operatorname { ln } ( k ) } }$

10-169.

For the graph of the equation $x^4 - xy = 2$ , write an equation for the line tangent to the curve at the point $(–1, 1)$ . Homework Help ✎

10-170.

Use your calculator to evaluate an integral representing the volume of the solid generated when the region bounded by $y = −(x + 3)^2 + 2$ and $y = 2^x$ is rotated about the line $y = 5$ . 10-170 HW eTool (Desmos) Homework Help ✎

10-171.

Express the arc length of each of the following curves as an integral, then evaluate the integral. Homework Help ✎

$f(y) = y^3$ from $y = -2$ to $y = 2$
$x = \frac { 2 } { 5 } ( y - 2 ) ^ { 5 / 2 }$ over $1 ≤ y ≤ 4$

10-172.

An object moves along the $x$ -axis with an initial position of $x(0) = 2$ . The velocity of the object when $t > 0$ is given by the equation $v ( t ) = 5 \operatorname { cos } ( \frac { \pi } { 2 } t ) + 4 t$ . 10-172 HW eTool (Desmos). Homework Help ✎

What is the acceleration of the object when $t = 4$ ?
What is the position of the object when $t = 4$ ?
What is the total distance the object travels over $0 ≤ t ≤ 4$ ?

10-173.

Suppose that $x = 9t^2$ and $y = \frac { 1 } { 2 - ( 3 t ) ^ { 2 } }$ . 10-173 HW eTool (Desmos). Homework Help ✎

Express $y$ as a function of $x$ .
Express $x$ as a function of $y$ .
How will the graph of the parametric equations above be different if $x = u$ and $y = \frac { 1 } { 2 - u }$ ?

Math Notes

Convergence Tests

Divergence Test: (or the $n$ ^th Term Test) If the limit of the terms of a series is not zero, then the series must diverge (have an infinite sum). For example, if $S = \displaystyle\sum _ { k = 0 } ^ { \infty } a _ { k }$ and $\lim\limits _ { k \rightarrow \infty } a _ { k } \neq 0$ , then $S$ diverges.

Alternating Series Test: A series such as $–2 + 1 –\frac { 2 } { 3 } + \frac { 1 } { 2 } - \frac { 2 } { 5 } + \frac { 1 } { 3 } \dots$ is called an alternating series because the sign of each term alternates between negative and positive.

$S = \displaystyle\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } a _ { n }$ is an example of an alternating series.

$S$ will converge (that is, have a finite sum) only when the absolute values of the terms goes to $0$ as $n → ∞$ .

That is, $S$ will converge when $\lim\limits _ { n \rightarrow \infty } a _ { n } = 0$ , or equivalently when $\lim\limits _ { n \rightarrow \infty } | a _ { n } | = 0$ .

Integral Test: Let $S = \displaystyle\sum _ { n = 1 } ^ { \infty } a _ { n }$ be an infinite series of decreasing terms.

If $f(n) = a_n$ and if $f$ is a continuous, positive, decreasing function over $[k, ∞)$ , then:

If $\int _ { k } ^ { \infty } f ( x ) d x$ converges, then $S$ converges.

If $\int _ { k } ^ { \infty } f ( x ) d x$ diverges, then $S$ diverges.

$\boldsymbol p$ -Series Test: The $p$ -series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { p } } = \frac { 1 } { 1 ^ { p } } + \frac { 1 } { 2 ^ { p } } + \frac { 1 } { 3 ^ { p } } + \ldots$ converges if $p > 1$ and diverges if $p ≤ 1$ .

Math Notes

Convergence Tests

Limit Comparison Test: Let $S = \displaystyle\sum _ { n = 1 } ^ { \infty } a _ { n }$ and $T = \displaystyle\sum _ { n = 1 } ^ { \infty } b _ { n }$ be two series where $a_n ≥ 0$ and $b_n > 0$ for all $n$ . If $\lim\limits _ { n \rightarrow \infty } \frac { a _ { n } } { b _ { n } } = r$ , and $r$ is finite and not zero, then $S$ and $T$ either both converge or both diverge.

If $r$ is infinite or zero, then the test is inconclusive and another test must be used.

Comparison Test: When dealing with series composed of positive terms, we can use a comparison to help decide if a series converges or diverges.

Suppose that $S$ and $T$ are infinite series with positive terms such that every term of $T$ is less than or equal to the corresponding term of $S$ . If $S$ converges then $T$ must also converge.

Likewise, suppose $S$ and $T$ are infinite series with positive terms such that every term of $T$ is greater than or equal to the corresponding term of $S$ . If $S$ diverges then $T$ must also diverge.

Given $0 ≤ a_n ≤ b_n$ for all $n$ :

If $\displaystyle\sum _ { n = 1 } ^ { \infty } b _ { n }$ converges then $\displaystyle\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges.

If $\displaystyle\sum _ { n = 1 } ^ { \infty } a _ { n }$ diverges, then $\displaystyle\sum _ { n = 1 } ^ { \infty } b _ { n }$ diverges.

Ratio Test: Let $S =\displaystyle\sum _ { n = 1 } ^ { \infty } a _ { n }$ be an infinite series such that $\lim\limits _ { k \rightarrow \infty } | \frac { a _ { k + 1 } } { a _ { k } } | = r$ .

Then: If $r < 1$ , the series converges.

If $r > 1$ , the series diverges.

If $r = 1$ , the test is inconclusive and another test must be tried.

Calculus Third Edition

Convergence Tests

Convergence Tests