Find the Taylor series for centered around .
I've gotten pretty close but I don't know how to put it back into summation format. This is my working:
So we have:
So I know the summation form must have but I can't find the pattern for
Help please, thanks!
A product of even numbers, 2*4*6*8*...*(2n) can be written as (2*1)(2*2)(2*3)(2*4)*...(2*n). Now factor out those "2"s: (2*2*2*2...2)(1*2*3*4...*n)= .
To simplify a product of odd numbers, 1*3*5*7*...*(2n-1), multiply and divide by the "missing" even numbers:
The numerator is now (2n)! and the denominator is .