# Power rule.

• February 5th 2010, 10:05 AM
BraveHeart
Power rule.
Prove that:

$\displaystyle{\left(\sum_{k=0}^{n}a_{k}x^{k}\right )' = \sum_{k=1}^{n}ka_{k}x^{k-1}}$

and/or that

$\int\left(\sum_{k=0}^{n}a_{k}x^{k}\right){dx} = \sum_{k=0}^{n}\frac{a_{k}x^{k+1}}{k+1}+c$
• February 5th 2010, 02:03 PM
TKHunny
I'm feeling an induction proof coming on. It's not like they are infinite sums. You won't have to worry about convergence.

Maybe I'm wrong.
• February 6th 2010, 04:21 AM
HallsofIvy
No, completely right. Induction on n and the "sum rule": (f(x)+ g(x))'= f'(x)+ g'(x).
• February 10th 2010, 09:20 PM
BraveHeart
I'm not sure how to proceed with induction.

Because $\left(\sum_{k=0}^{n}a_{k}x^{k}\right)' = \sum_{k=0}^{n}\left(a_{k}x^{k}\right)' = \sum_{k=0}^{n}a_{k}\left(x^{k}\right)'$, what needs to be proven is that $\left(x^{k}\right)' = kx^{k-1}$. The same goes to $\int\left(\sum_{k=0}^{n}a_{k}x^{k}\right){dx}
$
.