Question

How do I find the sum of an infinite series, like this one:

Answer 1

\[\sum_{n=0}^{\infty} (1-(-1)^n)/6^n\]

It converges to 12/35 but I feel like I'm doing it wrong because I'm just adding up the terms on paper until I get high enough that it looks about like it's approaching that.

Answer 2

write it as 2 geometric sums

Answer 3

\[\sum_{n=0}^{\infty}\frac{1}{6^n}-\sum_{n=0}^{\infty} \left(\frac{-1}{6}\right)^n\]

\[=\frac{6}{5}-\frac{6}{7}=\frac{12}{35}\]