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

.

Then evaluate