# Thread: help on taylor series (integral)

1. ## help on taylor series (integral)

f(x) = integral from 0 to x
((cos(t)-1)/(t^2))

I took the known taylor series of cos(t)
= Sigma n=0 to infinity
((-1)^n t^2n))
2n! t^2

and moved the t^2 to the t exponential
((-1)^n t^2n-2))
2n!

What would the next step be?
If I take the integral...
((-1)^n x^2n-1))
2n!(2n-1)
from n=1 to infinity

Is this correct?
Thanks.

2. Originally Posted by khuezy
f(x) = integral from 0 to x
((cos(t)-1)/(t^2))

I took the known taylor series of cos(t)
= Sigma n=0 to infinity
((-1)^n t^2n))
2n! t^2

and moved the t^2 to the t exponential
((-1)^n t^2n-2))
2n!

What would the next step be?
If I take the integral...
((-1)^n x^2n-1))
2n!(2n-1)
from n=1 to infinity

Is this correct?
Thanks.
Integrating:

$\displaystyle \sum_{n=0}^{\infty}(-1)^n \frac{t^{2n-2}}{(2n)!}$

term by term gives:

$\displaystyle \sum_{n=0}^{\infty}(-1)^n \frac{t^{2n-1}}{(2n)!(2n-1)}$

so that looks like a yes (other than for the lower limit of summation.

CB