Home > APCALC > Chapter 11 > Lesson 11.2.1 > Problem11-50

11-50.

Multiple Choice: The interval of convergence of the power series $\displaystyle\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { x ^ { n } } { n }$ is:

1. $(–∞, ∞)$

1. $[–1, 1]$

1. $[–1, 1)$

1. (–1, 1]

1. $\left(–1, 1\right)$

Use the Ratio Test.

$\lim\limits_{n\to\infty}\Bigg|\frac{\frac{x^{n+1}}{n+1}}{\frac{x^n}{n}}\Bigg|=$

$\lim\limits_{n\to\infty}|x|\cdot\frac{n}{n+1}=$

$|x|<1$

If $x = -1$:

$\displaystyle\sum_{n=1}^{\infty}(-1)^{n-1}\frac{(-1)^n}{n}=\sum_{n=1}^{\infty}\frac{(-1)^{2n-1}}{n}=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{(-1)}{n}$

Does this resultant series converge or diverge?
Be sure to check the resultant series if $x = 1$.