Good idea! I'm interested to know where this thread will get.
There used to be a time when mathematicians were champions of series, infinite products and special functions. If you couldn't give the series expansion of Jacobi's elliptic functions in a minute, there was no way you'd even be admitted to study at Oxford in 1900.
With this said, today most mathematicians don't care so much about series and special functions. This isn't to say that such things are less important, but they are less important in comparison to more modern mathematics. We still very much enjoy seeing the solution to a nice series, because it is pleasing to the eye and to the mind.
In this thread, I will post some kind of series or part of an identity, which has to be evaluated. Feel free to post some as well!
Let's begin with something not too hard. First, evaluate
Here I attempt to calculate this integral ( only when ) without using or even this fact
First make the substitution or , then The integral becomes :
These three integrals come from American Mathematical Monthly (AMM E3140 )
Show that :
After the substitution , we will have
, well , which is more obvious for us to think of .
I've got the solutions , later i will include them in this thread .