Sum to the infinity of a series

[justwild]

Homework Statement

to find the value of $\sum$ over n=1 to ∞ of [1/{1+(n-1)2}](1/3)$^{2+(n-1)2}$

Homework Equations

The Attempt at a Solution

I have tried to solve in the way the arithmetico-geometric series are solved and tried to bring it in the form of the expansion of ln(1+y), because the answer has logarithmic term in it.

 

[Curious3141]

Homework Statement

to find the value of $\sum$ over n=1 to ∞ of [1/{1+(n-1)2}](1/3)$^{2+(n-1)2}$

Homework Equations

The Attempt at a Solution

I have tried to solve in the way the arithmetico-geometric series are solved and tried to bring it in the form of the expansion of ln(1+y), because the answer has logarithmic term in it.

Hints:

Get rid of the (n-1) by starting the sum off from n = 0, that'll make things clearer.

Observe that ${(\frac{1}{3})}^{2 + 2n} =\frac{1}{3}.{(\frac{1}{3})}^{2n + 1}$

What's $\int_0^x t^{2n} dt$?