Question

how do you take the derivative of a summation? ie: question : find the derivative of the summation from 0 to infinity of (x^2n)/n!

Answer 1

any ideas?

Answer 2

You can recognize the function the series represents and then find the derivative of that function.

Answer 3

Or, you can write out the first few terms of the given series, take the derivative term-by-term, and try to find a pattern in order to write the terms as a series.

Answer 4

how do you recognize the function the series represents?

Answer 5

i mean i guess it could represent e^2x?

Answer 6

The power series for the exponential function \(f(x)=e^x\) is
\[e^x=\sum_{n=0}^\infty\frac{x^n}{n!}\]

Answer 7

Close, it's actually \(e^{x^2}\).

Answer 8

oh so the derivative of e^x^2 is... ummm 2x e^x^2?

Answer 9

Yes. And you can also write that as a power series.

Answer 10

and thats the answer for the derivative of the summation?

Answer 11

how do you do it as power series

Answer 12

or take it term by term

Answer 13

That depends on the scope of the question, but either series or function representation would probably be acceptable.

As for writing the derivative as a power series:
Since \(e^{x^2}=\displaystyle\sum_{n=0}^\infty\frac{x^{2n}}{n!},\) you have
\[\begin{align*}2xe^{2x^2}&=2x\displaystyle\sum_{n=0}^\infty\frac{x^{2n}}{n!}\\
&=\displaystyle2\sum_{n=0}^\infty\frac{x^{2n+1}}{n!}\end{align*}\]

Answer 14

ty

Answer 15

You're welcome.