10.1.8Are all series geometric in the end?

The Ratio Test

10-90.

Given $S = \displaystyle\sum _ { n = 0 } ^ { \infty } a r ^ { n }$ .

List the first four terms of $S$ . What type of series is $S$ ?
For what values of $r$ will $S$ converge?

10-91.

Let $T = \displaystyle\sum _ { k = 1 } ^ { \infty } a _ { k }$ , where each term $a _ { k } = \frac { k } { 2 ^ { k } }$ .

Expand series $T$ , showing the first four terms. Is this series geometric? Explain.
Consider the tests you know for convergence. Which, if any, can be used to determine if $T$ converges?
When all tests you know so far fail, you can pretend that the series, in the end, becomes geometric. In other words, evaluate the ratio of successive terms as the number of terms approaches infinity. Evaluate $\lim\limits _ { k \rightarrow \infty } | \frac { a _ { k + 1 } } { a _ { k } } |$ and call this limit $r$ , for limiting ratio.
Just like a geometric series, since $\lim\limits _ { k \rightarrow \infty } | \frac { a _ { k + 1 } } { a _ { k } } | = r$ , eventually, $a_{k + 1} ≈ ra_k$ . Use the value of $r$ you computed in part (c) to conclude whether $T$ converges or diverges? Explain your conclusion.

10-92.

The Ratio Test

Though only geometric series have a common ratio between successive terms, the ratio between successive terms as the number of terms approaches infinity can be a helpful way to determine if a non-geometric series will converge or diverge.

Using your work from problem 10-91, copy and complete the statement below to write a conjecture describing how the ratio between successive terms of an infinite series can be used to determine if the series converges or diverges.

The Ratio Test

Let $s=\displaystyle \sum _{n=1} ^\infty a_n$ such that $\lim \limits _{n \rightarrow \infty} |\frac{a_{n+1}}{a_n}|=r$

If _________ , then $S$ ___________ .

If __________, then $S$ ___________.

What if $r = 1$ ? In the case of infinite geometric series, a ratio of $1$ will guarantee divergence. But what about non-geometric series? What is the ratio as $n → ∞$ for each of the following two series? Use the results to explain why the Ratio Test is inconclusive when $r = 1$ .
1. $\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$
1. $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$
If you have not done so already, revise your Ratio Test conjecture so that $r = 1$ is correctly included.
If ____________ , then the Ratio Test is __________ .
$\$

10-93.

Revisit the series in problem 10-2. For which series could you have used the Ratio Test to determine convergence? For each, use either test to determine if the series converges.

10-94.

Test the each of following infinite series for convergence. State the tests you used.

$\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { k ! } { k ^ { 3 } }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { cos } ( \pi n ) } { n }$

$\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { k } { k ^ { 2 } + 1 }$

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

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { ln } ( n ) } { e ^ { n } }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } }$

10-95.

Let $\frac { d y } { d x } = 0.25 y ( 3 - y )$ . Homework Help ✎

Let $∆x = 0.5$ . Plot four different curves with four different starting points using Euler’s Method: $(0,1),(0,1.5),$ $(0,3),(0,5)$ . Plot points until you get to $x = 3$ .
What do you notice about all four curves? What is it about $\frac { d y } { d x }$ which causes this to happen?

10-96.

Examine the following series. Use one of the tests you have learned so far to determine if the series converges or diverges. State the tests you used. Homework Help ✎

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

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

$\displaystyle\sum _ { n = 1 } ^ { \infty } n e ^ { - n ^ { 2 } }$

10-97.

Write the equation of a line such that $f^\prime(3) = 2$ and $f(3) = 5$ . Now write the equation of the normal line at $x = 3$ . Homework Help ✎

10-98.

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 ✎

$\int \frac { \operatorname { sec } ^ { 2 } ( t ) } { 1 + \operatorname { tan } ( t ) } d t$

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

$\int_{ }^{ }\frac{d}{dx}(e^{\sqrt{2x}})dx$

$\int \frac { 1 } { x \sqrt { \operatorname { ln } ( x ) } } d x$

10-99.

What is the arc length of the curve $y = mx + b (m ≠ 0)$ from $x = k_l$ to $x = k_2 (k_l < k_2)$ ? Use the arc length formula. Then verify your result using the Pythagorean Theorem. Illustrate your work with a diagram. Homework Help ✎

10-100.

Sketch a possible graph for the following situation. Do not worry about units, but label the axes appropriately. Homework Help ✎

Archie arrives at school with an amazing rumor about his math teacher. He tells the first group of people he sees and then those people start telling others. Sketch a graph of the people who know the rumor as a function of time.

10-101.

Given the curve whose equation is $xy^4 + x^3y = -6$ : Homework Help ✎

Write an equation for $\frac { d y } { d x }$ .
What is the slope of the curve at $(2, –1)$ ?

10-102.

Examine the slope field shown at right. Imagine drawing some solutions, beginning with a variety of $y$ -intercepts above the $x$ -axis. What will be $\lim\limits _ { x \rightarrow - \infty } y ( x )$ for each of the solutions? Homework Help ✎

Coordinate plane, 10 rows of 10 short segments, each row with same slope at given y values, as follows, @ negative 2, slope of 6, @ 0.25, slope of negative 1 half, @ 2, slope of negative 2, @ 4.5, slope of negative 3, @ 6.5, slope of negative 4, @ 8.5, slope of negative 1, @ 10.5, slope of 4, @ 12.5, 14.5, & 16.5, each has almost vertical slope. Your teacher can provide you with a model.

Calculus Third Edition

The Ratio Test