# Determine whether the series converges, and if it converges, determine its value.

• May 18th 2010, 10:09 PM
ewkimchi
Determine whether the series converges, and if it converges, determine its value.
Consider the series Determine whether the series converges, and if it converges, determine its value.

• May 18th 2010, 10:31 PM
Prove It
Quote:

Originally Posted by ewkimchi
Consider the series Determine whether the series converges, and if it converges, determine its value.

$\displaystyle \sum_{n =1}^{\infty}\frac{(-7)^{n- 1}}{10^n} = \frac{1}{10} - \frac{7}{100} + \frac{49}{1000} - \frac{343}{10\,000} + \dots - \dots$.

This is a geometric series with $\displaystyle a = \frac{1}{10}$ and $\displaystyle r = -\frac{7}{10}$.

Since $\displaystyle |r| < 1$, the series is convergent, and

$\displaystyle S_{\infty} = \frac{a}{1 - r}$

$\displaystyle = \frac{\frac{1}{10}}{1 - \left(-\frac{7}{10}\right)}$

$\displaystyle = \frac{\frac{1}{10}}{\frac{17}{10}}$

$\displaystyle = \frac{1}{17}$.