# Maclaurin series expansion

• Oct 9th 2009, 03:51 AM
HD09
Maclaurin series expansion
I want to calculate the Maclaurin series for $\displaystyle log(1+x^2)$ by using a previous result, that $\displaystyle log(1+x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \frac{1}{4}x^4 + ...$

The following is not correct, but I don't know why, nor how to do it the right way

let $\displaystyle u=x^2$. Then

$\displaystyle log(1+x^2) = log(1+u) = u - \frac{1}{2}u^2 + \frac{1}{3}u^3 - \frac{1}{4}u^4 + ... = x^2 - \frac{1}{2}x^4 + \frac{1}{3}x^6 - \frac{1}{4}x^8 + ...$

Thanks for any help.
• Oct 9th 2009, 05:09 AM
Jester
Quote:

Originally Posted by HD09
I want to calculate the Maclaurin series for $\displaystyle log(1+x^2)$ by using a previous result, that $\displaystyle log(1+x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \frac{1}{4}x^4 + ...$

The following is not correct, but I don't know why, nor how to do it the right way

let $\displaystyle u=x^2$. Then

$\displaystyle log(1+x^2) = log(1+u) = u - \frac{1}{2}u^2 + \frac{1}{3}u^3 - \frac{1}{4}u^4 + ... = x^2 - \frac{1}{2}x^4 + \frac{1}{3}x^6 - \frac{1}{4}x^8 + ...$

Thanks for any help.

Why is that not correct (on [-1,1])?
• Oct 9th 2009, 06:24 AM
HD09
Because my solutions say so and the long-way calculation I did (which turns out to have a mistake in it now I check again) agreed with those instead of this!! Thanks. I was flabbergasted at how this could possibly be incorrect... what a relief.
• Oct 9th 2009, 06:35 AM
HallsofIvy
Now you've got me curious! What was the "solution sheet" answer that "the long way" agreed with?
• Oct 9th 2009, 07:00 AM
HD09
It only gave the first two terms but the solutions claim the co-efficient of the second term is 1/4 rather than 1/2