W10 Regular

Due date: Sunday 3/22, 11:59pm

Ratio Test and Root Test

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Ratio and root tests

Apply the ratio test or the root test to determine whether each of the following series is absolutely convergent, conditionally convergent, or divergent.

(a) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-2)}^n}{n^{100}}}$ $$ (b) ${\displaystyle \sum _{n=0}^{\infty }{\left(\frac{5n}{10n+4}\right)}^n}$ $$ (c) ${\displaystyle \sum _{n=1}^{\infty }\frac{\sqrt{n}}{3^n}}$

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Solution

Solutions---0200-02

Series tests: strategy tips

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Various limits, Part I

Find the limits. You may use $+\infty$ or $-\infty$ or $\mathrm{DNE}$ as appropriate. Braces indicate sequences.

• C = Convergent
• AC = Absolutely Convergent
• CC = Conditionally Convergent
• D = Divergent

$a_n$	${\lim }_{n\to \infty }{a}_n$	$\left\{a_n\right\}$ C or D	${\lim }_{n\to \infty }{(-1)}^n{a}_n$	$\left\{{(-1)}^n{a}_n\right\}$ C or D	$\sum a_n$ AC, CC, or D	$\sum {(-1)}^n{a}_n$ AC, CC, or D
${\displaystyle \frac{1}{n+2}}$
${\displaystyle \frac{n}{n+2}}$
${\displaystyle \frac{1}{n^2+2}}$
${\displaystyle \frac{4}{2^n}}$
${\displaystyle \frac{4n}{2^n}}$

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Solution

Solutions---0210-01

03 $★$

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Various limits, Part II

Find the limits. You may use $+\infty$ or $-\infty$ or $\mathrm{DNE}$ as appropriate. Braces indicate sequences.

• C = Convergent
• AC = Absolutely Convergent
• CC = Conditionally Convergent
• D = Divergent

$a_n$	${\lim }_{n\to \infty }{a}_n$	$\left\{a_n\right\}$ C or D	${\lim }_{n\to \infty }{(-1)}^n{a}_n$	$\left\{{(-1)}^n{a}_n\right\}$ C or D	$\sum a_n$ AC, CC, or D	$\sum {(-1)}^n{a}_n$ AC, CC, or D
${\displaystyle \frac{4n!}{2^n}}$
${\displaystyle \frac{(n+2)3^n}{n!}}$
${\displaystyle \frac{4^n}{{(3n)}^n}}$
${\displaystyle \frac{1}{(2n+1)!}}$

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Solution

Solutions---0210-02

Power series: Radius and Interval

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Power series - radius and interval

Find the radius and interval of convergence for these power series:

(a) ${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\frac{{(x+3)}^n}{n!}}$ $$ (b) ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^n\frac{{(x-7)}^n}{n}}$ $$ (c) ${\displaystyle \sum _{n=12}^{\infty }{n}^n{(x-2)}^n}$

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Solution

Solutions---0220-02

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Power series - radius and interval

Find the radius and interval of convergence for these power series:

(a) ${\displaystyle \sum _{n=0}^{\infty }\frac{{(x-8)}^n}{n^4+1}}$ $$ (b) ${\displaystyle \sum _{n=1}^{\infty }\frac{x^n}{3\cdot 7\cdot 11\cdots (4n-1)}}$

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Solution

Solutions---0220-03