Is it really homework? Then you better NOT submit this:

use strict;
use warnings;
use bignum accuracy => 30;
use feature 'say';
$/=--$,;say+4*($\+=($/*=-1)/($,+=2))for+1..10;
[download]

This also sounds like homework. Please show what you have tried.

Also, please note that this should have been posted in SoPW, not meditations. It has been moved.

That was also my first reaction, it really looks like homework.

I remember when I was in college one semester my calculus instructor (who was also the computer science instructor) gave us a scenario in which we had to compute how many steps we needed to perform using the same (Taylor?) series, to generate the value for a particular accuracy. (In one of my computer science classes that same semester {in either Fortran or Pascal-memory fails me now as to which}, she had us implement it.)

Thanks, OP, for a reminder of some good times. :)

you need to implement your own tricky iterative algorithm using pure integer arithmetics. Good luck!

Math::BigInt may be helpful for this.