Sum of series of real numbers. I am trying to find the sum of the series $$1+\frac{1}{3.4}+\frac{1}{5.4^{2}}+\frac{1}{7.4^{3}}\cdot\cdot\cdot$$ What is the idea behind to find the sum of this type of series? How i think to compare it to other known series like log series etc.? Please help. Thanks in advance.

Recall the MacLaurin series for $\ln(1+x)$,
$$x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots,$$ 
and the series for $\ln(1-x)$, 
$$x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\cdots.$$
Add, divide by $2$. We get
$$\frac{1}{2}(\ln(1+x)+\ln(1-x))=x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots.$$
Divide by $x$, and make the appropriate choice for $x$.
