Calculating the Value of an Infinite Series

[tommyhakinen]

Homework Statement

What is the value of:
1 + $$(\frac{1}{3})^{2}$$ + $$(\frac{1}{5})^{2}$$ + $$(\frac{1}{7})^{2}$$ + $$(\frac{1}{9})^{2}$$ + ...

 

[Tedjn]

Use the following:

$$\sum_{i=1}^\infty \left(\frac{1}{2i}\right)^2 = \frac{1}{4}\sum_{i=1}^\infty \frac{1}{i^2}.$$​

You will need to rely on the fact that the series converges absolutely. (Why?)

 

[chaoseverlasting]

Use the following:

$$\sum_{i=1}^\infty \left(\frac{1}{2i}\right)^2 = \frac{1}{4}\sum_{i=1}^\infty \frac{1}{i^2}.$$​

You will need to rely on the fact that the series converges absolutely. (Why?)

That won't work. The series in question has squares of odd numbers in the denominator.

Let the given series be S. Now, let us assume that there is another series K which is the sum of all the inverse squares of the even numbers till infinity.

Now, $$S+K=\sum \frac{1}{i^2}$$

This series K is the series that Tedgin pointed out. You can calculate S+K, and K. Subtract the two and you get your answer.