Practice Chart - Solutions

You may use $+\infty$ or $-\infty$ or $\mathrm{DNE}$ as appropriate. Curly braces indicate sequences.

Notations:

• 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
${\displaystyle \frac{2n}{n+2}}$	2	$C$	$DNE$	$D$	$D$	$D$
${\displaystyle \frac{3^n}{n^2}}$	$\infty$	$D$	$DNE$	$D$	$D$	$D$
${\displaystyle {\left(\frac{2n+8}{n^2}\right)}^n}$	0	$C$	0	$C$	$AC$	$AC$
${\displaystyle \frac{n}{n^2+3}}$	0	$C$	0	$C$	$D$	$CC$
${\displaystyle \frac{n!}{{100}^n}}$	$\infty$	$D$	$DNE$	$D$	$D$	$D$
${\displaystyle \frac{1}{1+\ln n}}$	0	$C$	0	$C$	$D$	$CC$
${\displaystyle 5{\left(\frac{3}{4}\right)}^n}$	0	$C$	0	$C$	$AC$	$AC$
${\displaystyle \frac{{10}^n}{n!}}$	0	$C$	0	$C$	$AC$	$AC$