CPM Homework Help : APCALC Problem 10-107

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10-107.

Decide if each of the following series converges or diverges. State the tests you used.

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n } { 2 n + 5 }$
Hint (a):
$\text{Is }\lim\limits_{n\to\infty}\frac{n}{2n+5}=0?$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n ! } { 5 ^ { n } }$
Hint (b):
Use the Ratio Test.

$\displaystyle\sum _ { n = 1 } ^ { \infty } \operatorname { ln } ( \frac { 1 } { n } )$
Hint (c):
$\text{Is }\lim\limits_{n\to\infty}\ln\Big(\frac{1}{n}\Big)=0?$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { cos } ( n ) } { 2 ^ { n } }$
Hint (d):
$\left|\cos(n)\right|\le1$
More Help (d):
$\frac{\cos(n)}{2^n}\le\frac{1}{2^n}$

$\displaystyle\sum _ { n = 0 } ^ { \infty } \frac { ( - 5 ) ^ { n } } { 4 ^ { n } }$
Hint (e):
This is a geometric series. What is the value of $r$ ?

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 5 } { n ^ { 4 } + 6 }$
Hint (f):
Use the Limit Comparison Test with $1/n^4$ .

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