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

**ewkimchi** · May 18th 2010, 10:09 PM

Consider the series

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

I know it converges, I got 7/10, but it's wrong. Please help! =)

**Prove It** · May 18th 2010, 10:31 PM

Originally Posted by ewkimchi

Consider the series

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

I know it converges, I got 7/10, but it's wrong. Please help! =)

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

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

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

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

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

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

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

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