TITLE: Calculate $\frac13+\frac29+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\ldots$
QUESTION [10 upvotes]: How can we calculate the formula for the following series?
$\frac13+\frac29+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\ldots$
I'm not quite sure what type of series it is anyway. It doesn't look geometric. Is there any strategy/procedure to follow in order to calculate the formula for a series of numbers?

REPLY [10 votes]: If we add neighbouring pairs, we get
$$\frac{5}{9}+\frac{5}{81}+\frac{5}{729}+\cdots.$$
This is an infinite geometric series with first term $\frac{5}{9}$ and common ratio $\frac{1}{9}$.
By the usual formula for the sum of an infinite geometric progression, the sum is 
$$\frac{5}{9}\frac{1}{1-\frac{1}{9}}.$$
This can be simplified to $\frac{5}{8}$.