CPM Homework Help : APCALC Problem 12-106

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12-106.

Consider the infinite series below. For each series, decide if it converges conditionally, converges absolutely, or diverges and justify your conclusion. State the tests you used.

$\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { 1 + \operatorname { cos } ( n ) } { n ^ { 2 } }$
Hint (a):
Compare this series to $\displaystyle \sum_{n=1}^\infty \frac{2}{n^2}.$

$\displaystyle\sum _ { j = 1 } ^ { \infty } \frac { j ! } { ( 2 j + 1 ) ! }$
Step (b):
$\lim\limits_{j\to\infty}\Bigg|\frac{\frac{(j+1)!}{(2j+2)!}}{\frac{j!}{(2j+1)!}}\Bigg|$

$\displaystyle\sum _ { n = 1 } ^ { \infty } n e ^ { - n ^ { 2 } }$
Step (c):
$\lim\limits_{n\to\infty}\Bigg|\frac{\frac{n+1}{e^{(n+1)^2}}}{\frac{n}{e^{n^2}}}\Bigg|$

$\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { 2 - k } { k \cdot 2 ^ { k } }$
Step (d):
$\lim\limits_{k\to\infty}\Bigg|\frac{\frac{1-k}{(k+1)\cdot 2^{k+1}}}{\frac{2-k}{k\cdot 2^k}}\Bigg|$

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