Infinite Series Summation of Fractions #1

As a general rule, $\displaystyle\sum_{n\ =\ 1}^{\infty}{1\over k^{n}} =\ ?$ should always equal $1\over {k-1}$ . Therefore the answer is ${1\over 7} \approx 0.143$ . Check this website for more information and a visual guide to a similar problem.

Another solution:

Take $\displaystyle\sum_{n\ =\ 1}^{\infty}{1\over k^{n}} = n$ . $8n = 1+\displaystyle\sum_{n\ =\ 1}^{\infty}{1\over k^{n}}$ . Therefore, $7n = (1+\displaystyle\sum_{n\ =\ 1}^{\infty}{1\over k^{n}}) - (\sum_{n\ =\ 1}^{\infty}{1\over k^{n}}) = 1$ , $n = {1\over 7} \approx 0.143$