W10 Regular

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

01

02

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

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02 $★$

01

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
$\frac{1}{n+2}$
$\frac{n}{n+2}$
$\frac{1}{n^2+2}$
$\frac{4}{2^n}$
$\frac{4n}{2^n}$

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03 $★$

02

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
$\frac{4n!}{2^n}$
$\frac{(n+2)3^n}{n!}$
$\frac{4^n}{(3n{)}^n}$
$\frac{1}{(2n+1)!}$

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04 $★$

02

Power series - radius and interval

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

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

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05

03

Power series - radius and interval

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

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

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