Double Summation Result: \alpha^i\alpha^j

[feryee]

what is the result for the following double summation:
$\sum\limits_{i \neq j}^{\infty}\alpha^i\alpha^j$

where $i, j =0,1,2,...$

 

[micromass]

Express it as an iterated sum and then apply the result for a sum of a geometric series. What do you get?

 

[feryee]

Express it as an iterated sum and then apply the result for a sum of a geometric series. What do you get?

Well actually i have the final result but simply i couldn't get the same answer using geometric sum. Here is the final result:
$\frac{2\alpha}{(1+\alpha)(1-\alpha)^2}$

How is it possible.

 

[micromass]

Have you tried anything? Where are you stuck?