10.1.6Mirror, mirror, on the wall, who's the smallest of them all?

The Comparison Test

10-63.

Examine the series below and determine if they converge or diverge, then give a reason why.

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

$S = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$

10-64.

Some series look very similar. Consider $S = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ and $T = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + 1 }$ .

List the first four terms of $S$ and $T$ .
Compare corresponding terms of $S$ and $T$ and plot them on the same set of axes. What do you notice?
What can you conclude about the convergence of $S$ ? What can you conclude about the convergence of $T$ ? Explain your conclusions.

10-65.

Now consider $Q = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ and $R = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n - 0.5 }$ .

List the first four terms of $Q$ and $R$ .
Compare corresponding terms of $Q$ and $R$ and plot them on the same set of axes. What do you notice?
What can you conclude about the convergence of $Q$ ? What can you conclude about the convergence of $R$ ? Explain your conclusions.

10-66.

The Comparison Test

Copy and complete the statement below to write a conjecture describing how you can compare the terms of an unfamiliar series $T$ with the corresponding terms of a familiar series $S$ to determine if they both converge or both diverge.

The Comparison Test

Given $S=\displaystyle \sum_{n=1} ^\infty a_n$ and $T=\displaystyle \sum_{n=1} ^\infty b_n$ where $O \leq a_n \leq b_n$ for all $n$ , then:
If _______ converges, then ________ converges.
If __________ diverges, then ________ diverges.

10-67.

Create your own example for each of the following situations.

By comparison with the convergent series $T = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 } }$ , a series $S$ can be shown to converge. Create a possible series $S$ . Support your answer by graphing the terms of both series on the same set of axes.
By comparison with the divergent series $W = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \sqrt { n } }$ , a series $Z$ can be shown to diverge. Create a possible series $Z$ . Support your answer by graphing the terms of both series on the same set of axes.

10-68.

Decide if each of the following series converges or diverges. Justify your answers, including which tests you used.

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

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

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

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

10-69.

Decide if each of the following series converges or diverges. Justify your answers, including which tests you used. Homework Help ✎

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

$\displaystyle\sum _ { n = 1 } ^ { \infty } n ^ { - 2001 }$

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

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { tan } ^ { - 1 } ( n ) } { n ^ { 2 } + 1 }$

10-70.

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 ✎

$\int \frac { 1 } { ( x - 1 ) ( x + 5 ) } d x$

$\int \frac { x ^ { 3 } + 2 x } { x - 1 } d x$

$\int 5 x ^ { 2 } ( 5 x ^ { 3 } - 2 ) ^ { - 1 / 2 } d x$

$- \int 2 \operatorname { sec } ^ { 2 } ( x ) d x$

10-71.

Determine all points of intersection for $r = 5\sin(θ)$ and $r = 2 + \sin(θ)$ for $0 ≤ θ ≤ 2π$ . Homework Help ✎

10-72.

For $2 ≤ x ≤ 4$ , determine the point(s) on the graph of $y = x^4 - 12x^3 + 52x^2 - 96x + 64$ at which the slope is the steepest. Homework Help ✎

10-73.

Write an integral to represent the volume of the solid created when the region bounded by $y = (x - 1)^2$ and $y =\sqrt { x }$ is rotated about the line $y = -1$ . Homework Help ✎

10-74.

Mariah walks towards a wall with a flashlight beaming straight ahead. When she is $5$ feet away from the wall, the radius of the circle of light is $2$ feet. If she walks at a rate of $0.5$ ft/sec, calculate the rate at which the diameter of the circle of light is changing. Homework Help ✎

10-75.

If $\frac { d y } { d x } = | x |$ and $(−2, 3)$ is a point on the solution curve, use Euler’s Method to sketch $y$ for $−2 ≤ x ≤ 2$ . Use $∆x = 0.5$ . Homework Help ✎

Calculus Third Edition

The Comparison Test