### Home > APCALC > Chapter 10 > Lesson 10.2.2 > Problem10-120

10-120.

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

1. $\displaystyle \sum _ { n = 1 } ^ { \infty } n ( \frac { 2 } { 3 } ) ^ { n }$

The Ratio Test can be used.

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

$n+\sqrt{n}\le 2n$

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

$\ln\Big(\frac{n}{n+1}\Big)=\ln(n)-\ln(n+1)$

Write out some terms of the series. You should notice that this is a telescoping series.

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

What is the value of the first term of the series?