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

#### ewkimchi

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

#### Prove It

MHF Helper
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

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

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

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

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