1. ## MacLaurin Series

I know how to do the easy maclaurin series questions like;

f(x) = 1/(2+x)

but im a bit clueless on the more complicated ones like;

(i) sin^x2

(ii) coshx = (e^x + e^-x)/2

(iii) integral from 0 to X of e^t2dt.

Any help would be greatly appreciated

2. Originally Posted by SirOJ
I know how to do the easy maclaurin series questions like;

f(x) = 1/(2+x)

but im a bit clueless on the more complicated ones like;

(i) sin^x2

(ii) coshx = (e^x + e^-x)/2

(iii) integral from 0 to X of e^t2dt.

Any help would be greatly appreciated
(i) Use $\sin^2 x = \frac{1}{2} \left( 1 - \cos(2x) \right)$.

(ii) There is a simple pattern to the derivatives of cosh(x) evaluated at x = 0.

(iii) Substitute $x = t^2$ into the Maclaurin series for $e^x$ and integrate the result.

3. (ii) Or, if you already know that $e^x= \sum_{n=0}^\infty \frac{x^n}{n!}$ $= 1+ x+ \frac{1}{2}x^2+ \frac{1}{6}x^3+ \cdot\cdot\cdot$, then you also know that $e^{-x}= \sum_{n=0}^\infty \frac{(-x)^n}{n!}$ $= 1- x+ \frac{1}{2}x^2- \frac{1}{6}x^3+ \cdot\cdot\cdot$ so that $cosh(x)= \frac{e^x+ e^{-x}}{2}= 1+ \frac{1}{2}x^2+ \frac{1}{4!}x^4+ \cdot\cdot\cdot$ $= \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}$

4. thanks alot guys, it's all clear to me now