Are you familiar with mathematical induction?
Try taking advantage of conjugates. For example, the first term can be rewritten .
In each case, the denominator is 2, so we can factor out a half.
The series eventually becomes .
Does this look nicer? Good luck.
I've been racking my head all day to do this without induction, i'm curious if anyone has an answer. I took a look at trying to do this with continued fractions but i had no idea what i was doing, if only i was ramanujan.
If I were going to use induction, I would state the hypothesis:
base case :
Now, adding this to the hypothesis, we have:
We have derived from , thereby completing the proof by induction, and we may now state:
I don't know why i didn't think of this earlier.
But if you have then
So after computing the integral i got , and after evaluating i got