# nth derivative taylor series

• Apr 21st 2008, 08:28 AM
polymerase
nth derivative taylor series
Let $\displaystyle f^{(n)}(a)$ denote the n-th derivative of $\displaystyle f$ at $\displaystyle a$. if $\displaystyle f(x)=xe^{2x}$, then $\displaystyle f^{(64)}(0)=$

I can get the answer using the old fashion way, but when I lookeed at the solution using power series, I don't understand one of the steps. heres the soultion my prof has:

$\displaystyle e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$
$\displaystyle f(x)=xe^{2x}=\sum_{n=0}^\infty \frac{2^nx^{n+1}}{n!}=\sum_{m=1}^\infty \frac{2^{m-1}x^{m}}{(m-1)!}= \boxed{\sum_{m=1}^\infty \frac{f^{(m)}(0)}{m!}x^m}$

hence, $\displaystyle f^{(m)}(0)=\frac{2^{m-1}m!}{(m-1)!}$

I understand every step except the step to the result in the box. How are they equal?
• Apr 21st 2008, 08:30 AM
Moo
Hello,

The thing in the box is the result of the Taylor series : Taylor series - Wikipedia, the free encyclopedia
It is said that the sum begins at m=1, but here it doesn't matter as f(0)=0

:)
• Apr 21st 2008, 08:42 AM
polymerase
Quote:

Originally Posted by Moo
Hello,

The thing in the box is the result of the Taylor series : Taylor series - Wikipedia, the free encyclopedia
It is said that the sum begins at m=1, but here it doesn't matter as f(0)=0

:)

nvm i figured it out...
What I was really trying to get at was how does $\displaystyle \sum_{m=1}^\infty \frac{f^{(m)}(0)}{m!}x^m=\sum_{n=0}^\infty \frac{2^nx^{n+1}}{n!}$
• Apr 21st 2008, 08:44 AM
Moo
nvm ?

The left hand part comes from the Taylor series
The right hand part comes from the power series :-)

$\displaystyle e^x=\sum_{n \geq 0} \frac{x^n}{n!}$

Hence $\displaystyle e^{2x}=\sum_{n \geq 0} \frac{(2x)^n}{n!}=\sum_{n \geq 0} \frac{2^n \ x^n}{n!}$

-> $\displaystyle xe^{2x}=\sum_{n \geq 0} \frac{2^n \ x^{n+1}}{n!}$
• Apr 21st 2008, 09:20 AM
polymerase
Quote:

Originally Posted by Moo
nvm ?

never mind :D