How do I compute $ \sum\limits_{n=0}^{\infty} \frac{1}{(n+2)7^n} $? I'm trying to evaluate the following sum:
$$ \sum\limits_{n=0}^{\infty} \frac{1}{(n+2)7^n} $$
Probably the best way to do that is to try and find a closed formula for 
$$ \sum\limits_{n=0}^{\infty}  \frac{1}{(n+2)x^n} $$
Probably it requires some integrating/differentiating. I've tried both, but I can't get it to look any more decently.
I'd appreciate some hint

$$\sum_{n\geq 0}\frac{1}{(n+2)7^n}=49\int_{0}^{1/7}\sum_{n\geq 0}x^{n+1}\,dx = 49\int_{0}^{1/7}\frac{x}{1-x}\,dx = \color{red}{-7+49\log\frac{7}{6}}.$$
