### Home > APCALC > Chapter 10 > Lesson 10.2.1 > Problem10-107

10-107.

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

1. $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n } { 2 n + 5 }$

$\text{Is }\lim\limits_{n\to\infty}\frac{n}{2n+5}=0?$

1. $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n ! } { 5 ^ { n } }$

Use the Ratio Test.

1.  $\displaystyle\sum _ { n = 1 } ^ { \infty } \operatorname { ln } ( \frac { 1 } { n } )$

$\text{Is }\lim\limits_{n\to\infty}\ln\Big(\frac{1}{n}\Big)=0?$

1. $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { cos } ( n ) } { 2 ^ { n } }$

$\left|\cos(n)\right|\le1$

$\frac{\cos(n)}{2^n}\le\frac{1}{2^n}$

1. $\displaystyle\sum _ { n = 0 } ^ { \infty } \frac { ( - 5 ) ^ { n } } { 4 ^ { n } }$

This is a geometric series. What is the value of $r$?

1. $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 5 } { n ^ { 4 } + 6 }$

Use the Limit Comparison Test with $1/n^4$.