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

Apr 2010
57
1
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

MHF Helper
Aug 2008
12,883
4,999
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! =)
\(\displaystyle \sum_{n =1}^{\infty}\frac{(-7)^{n- 1}}{10^n} = \frac{1}{10} - \frac{7}{100} + \frac{49}{1000} - \frac{343}{10\,000} + \dots - \dots\).


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

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

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

\(\displaystyle = \frac{\frac{1}{10}}{\frac{17}{10}}\)

\(\displaystyle = \frac{1}{17}\).