Problems

01

Various limits, Part I

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

C = Convergent

AC = Absolutely Convergent

CC = Conditionally Convergent

D = Divergent

$a_{n}$ $\lim_{n \to \infty} a_{n}$ ${a_{n}}$
C or D $\lim_{n \to \infty} {(- 1)}^{n} a_{n}$ ${{(- 1)}^{n} a_{n}}$
C or D $\sum a_{n}$
AC, CC, or D $\sum {(- 1)}^{n} a_{n}$
AC, CC, or D
$\frac{1}{n + 2}$
$\frac{n}{n + 2}$
$\frac{1}{n^{2} + 2}$
$\frac{4}{2^{n}}$
$\frac{4 n}{2^{n}}$

02

Various limits, Part II

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

C = Convergent

AC = Absolutely Convergent

CC = Conditionally Convergent

D = Divergent

$a_{n}$ $\lim_{n \to \infty} a_{n}$ ${a_{n}}$
C or D $\lim_{n \to \infty} {(- 1)}^{n} a_{n}$ ${{(- 1)}^{n} a_{n}}$
C or D $\sum a_{n}$
AC, CC, or D $\sum {(- 1)}^{n} a_{n}$
AC, CC, or D
$\frac{4 n!}{2^{n}}$
$\frac{(n + 2) 3^{n}}{n!}$
$\frac{4^{n}}{{(3 n)}^{n}}$
$\frac{1}{(2 n + 1)!}$

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$a_{n}$	$\lim_{n \to \infty} a_{n}$	${a_{n}}$ C or D	$\lim_{n \to \infty} {(- 1)}^{n} a_{n}$	${{(- 1)}^{n} a_{n}}$ C or D	$\sum a_{n}$ AC, CC, or D	$\sum {(- 1)}^{n} a_{n}$ AC, CC, or D
$\frac{1}{n + 2}$
$\frac{n}{n + 2}$
$\frac{1}{n^{2} + 2}$
$\frac{4}{2^{n}}$
$\frac{4 n}{2^{n}}$

$a_{n}$	$\lim_{n \to \infty} a_{n}$	${a_{n}}$ C or D	$\lim_{n \to \infty} {(- 1)}^{n} a_{n}$	${{(- 1)}^{n} a_{n}}$ C or D	$\sum a_{n}$ AC, CC, or D	$\sum {(- 1)}^{n} a_{n}$ AC, CC, or D
$\frac{4 n!}{2^{n}}$
$\frac{(n + 2) 3^{n}}{n!}$
$\frac{4^{n}}{{(3 n)}^{n}}$
$\frac{1}{(2 n + 1)!}$

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