SOLUTION: find the sum of the series 5/13 + 55/(13)^2 + 555/(13)^3 + ...

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Question 328453: find the sum of the series 5/13 + 55/(13)^2 + 555/(13)^3 + ...
Answer by galactus(183) About Me  (Show Source):
You can put this solution on YOUR website!
This is kind of tricky trying to set up a general form for 5,55,555,....
We can think of the numerators: 5,55,555,5555,...., as
%285%2F9%29%2810%5En-1%29
But, this gives us a general form of
%285%2F9%29%2Asum%28%2810%5En-1%29%2F13%5En%2Cn=1%2Cinfinity%29

Using the formula for the sum of an infinite series we get
%2810%2F13%29%2F%281-10%2F13%29=10%2F3
%281%2F13%29%2F%281-1%2F13%29=1%2F12
5%2F9%2A%2810%2F3-1%2F12%29=65%2F36
The sum is 65/36