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

1. ## 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.

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

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

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

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

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

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

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

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