10.4.1Even better than convergence!

Absolute Convergence

10-149.

Jack and his sister Anna are arguing about whether or not the following series converges: $\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { ( - 20 ) ^ { k } } { - ( k + 1 ) ! }$

Jack thinks they should use the Alternating Series Test to see if it converges and Anna thinks they should use the Ratio Test.

Use one of the tests you have learned to decide if the series converges or not. State the test that you used.

10-150.

Jack’s other sister Makayla has already taken Calculus. She says, “If you are having trouble with the negative signs, just take the absolute value of the terms, and then decide if it converges using one of your tests.”

Try Makayla’s method on the series $\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { ( - 20 ) ^ { k } } { - ( k + 1 ) ! }$ to determine if it converges. In other words, determine if $\displaystyle\sum _ { k = 1 } ^ { \infty } \Big{|} \frac { ( - 20 ) ^ { k } } { - ( k + 1 ) ! } \Big{|}$ converges.
Do you think Makayla’s method always works? Why or why not?

10-151.

Now try Makayla’s method on the alternating harmonic series $\displaystyle \sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k + 1 } \frac { 1 } { k }$ .

Determine if $\displaystyle \sum _ { k = 1 } ^ { \infty } \Big{|} ( - 1 ) ^ { k + 1 } \frac { 1 } { k } \Big{|}$ will converge.
Determine if the original series $\displaystyle\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k + 1 } \frac { 1 } { k }$ converges. State the tests you used.

10-152.

In this lesson we have studied two series, $\displaystyle \sum _ { k = 1 } ^ { \infty } \frac { ( - 20 ) ^ { k } } { - ( k + 1 ) ! }$ and $\displaystyle\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k + 1 } \frac { 1 } { k }$ . One of these series converges absolutely, and the other converges conditionally. In your own words, describe what converges conditionally means.

10-153.

Determine if each of the following series converges absolutely, converges conditionally, or diverges. Be sure to indicate which convergence test you used.

$\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { \sqrt [ 3 ] { k ^ { 2 } } }$

$\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { ( - 5 ) ^ { k } } { k ^ { 3 } }$

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

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

10-154.

Determine if each of the following series converges absolutely, converges conditionally, or diverges. State the tests you used. Homework Help ✎

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

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

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

$\displaystyle\sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } ( k + 1 ) ! } { 2 ^ { 4 k } }$

10-155.

For each of the geometric series state the common ratio “ $r$ ” and determine the sum of the series. Homework Help ✎

$2.5 + 0.75 + 0.225 + 0.0675 + …$

$\displaystyle\sum _ { k = 1 } ^ { \infty } 4 ( \frac { - 3 } { 5 } ) ^ { k }$

$\frac { 2 } { 15 } + \frac { 1 } { 5 } + \frac { 3 } { 10 } + \frac { 9 } { 20 } + \ldots$

$- 3 + \frac { 12 } { 5 } - \frac { 48 } { 25 } + \frac { 192 } { 125 } -\ldots$

10-156.

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

$\int _ { - \infty } ^ { \infty } \frac { 1 } { x ^ { 2 } + 25 } d x$

$\int _ { 0 } ^ { 4 } \frac { x } { \sqrt { 16 - x ^ { 2 } } } d x$

$\int _ { - \infty } ^ { - 1 } \frac { 1 } { x ^ { 3 } } d x$

$\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 2 x - 3 } d x$

10-157.

Meg and Jack are racing their frogs Dwayne and Jerry. 10-157 HW eTool (Desmos). Homework Help ✎

If Dwayne’s velocity in feet per second during the race is given by the equation $v(t)=-0.5(t+2\sin t-11)^2+6.05$ , determine Dwayne’s average velocity during $0 ≤ t ≤ 18$ . Describe your method.
If Jerry’s distance from the starting line during the race is given by the equation $s(t) = 1.5x + 0.2\sin(x^2)$ , determine Jerry’s average velocity during $0 ≤ t ≤ 18$ . Describe your method.
When is each frog traveling at his/her average velocity?

10-158.

Describe the difference between the graphs of a cardioid and a limaçon. Provide an example equation for each.10-158 HW eTool (Desmos) Homework Help ✎

10-159.

According to Newton’s law of cooling, the rate at which an object cools (or warms) is directly proportional to the temperature difference between the environment and the object itself. Three years ago the corpse of Dr. Deadman was discovered in the coroner’s office. The room temperature of the coroner’s office was ( $17^\circ \text{C}$ ). The doctor’s body temperature was measured to be $27^\circ \text{C}$ , which was $10^\circ \text{C}$ below normal. Homework Help ✎

Dr. Deadman’s body was found at 5:05 p.m. An hour later the body had cooled to $26^\circ \text{C}$ . At what rate was the body cooling when it was found?
Approximately when was Dr. Deadman killed?

10-160.

Calculate each of the following limits. Homework Help ✎

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

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

Hint: Use natural log.

$\lim\limits _ { x \rightarrow \infty } \frac { e ^ { x } } { x ^ { 2 } }$

$\lim\limits _ { x \rightarrow \infty } \frac { x } { \sqrt { x ^ { 2 } + 1 } }$

Calculus Third Edition

Absolute Convergence