### Home > APCALC > Chapter 10 > Lesson 10.1.6 > Problem10-69

10-69.

Decide if each of the following series converges or diverges. Justify your answers, including which tests you used.

1. $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n } { 3 ^ { n } ( n + 1 ) }$

Compare this series to $\displaystyle\sum_{n=1}^\infty \frac{1}{3^n}.$

1. $\displaystyle\sum _ { n = 1 } ^ { \infty } n ^ { - 2001 }$

This is a $p$-Series.

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

Compare this series to $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}.$

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

$\left|\tan^{-1}(n)\right|<\frac{\pi}{2}$

$\frac{1}{n^2+1}<\frac{1}{n^2}$