Good idea! I'm interested to know where this thread will get.
Spoiler:
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
So the answer to the original problem is
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 )
Let
Show that :
All of the other solutions in this thread, while ingenious, are relatively straightforward. However you, simplependulum, always come up with an elementary and magical substitution which does the trick! Where do you find these ideas? First, partial fractions backwards, and then the innocent-looking but not obvious ... Please share some of your techniques for finding good substitutions (other than "practice", obviously )!
For the first step to evaluate the integral , we can actually use the substitution followed by partial fractions , it becomes more obvious if we do this :
After the substitution , we will have
, well , which is more obvious for us to think of .
Talking about these three integrals (actually two , the ideas of the last two integrals are almost the same ) , I remember I'd spent a few weeks to finish when i was a fool who didn't realize the power of integration of parameter . Now , I know some basics of the techniques and these integrals look much easier to me BUT i have to say they are still difficult if we compare with many other integrals !!
I've got the solutions , later i will include them in this thread .