I went down a rabbit hole while reading An Imaginary Tale by Paul J Nahin and I decided I wanted to do this…
The power series expansion of is
Hence
Let’s consider the integral
(1)
Let then and where
When and when
(2)
(3)
Integrate by parts using the tabular method.
Sign | Differentiate | Integrate |
+ | ||
– | ||
+ | ||
– | ||
+ | ||
When we substitute or the differentiation column is zero except for , which is ,
Thus
Now we just need to think about the sign.
The integral is now
So
Let’s work out some partial sums