### Home > CALC > Chapter 10 > Lesson 10.2.2 > Problem10-103

10-103.

Given the series $S =\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { a ^ { n } } { n + 1 }$, where $a$ is a constant.

1. Does $S$ converge if $a = 1$? Justify your answer.

This is a variation of the harmonic series.

2. Does $S$ converge if $a = −1$? Justify your answer.

This is an alternating series.

3. For what values of $a$ does $S$ converge? Justify your answer.

Using the Ratio Test: $\lim\limits_{n\to\infty}\Big|\frac{a^{n+1}/(n+2)}{a^n/(n+1)}\Big|<1$