# Math Help - Power rule.

1. ## 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

$\int\left(\sum_{k=0}^{n}a_{k}x^{k}\right){dx} = \sum_{k=0}^{n}\frac{a_{k}x^{k+1}}{k+1}+c$

2. 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.

3. No, completely right. Induction on n and the "sum rule": (f(x)+ g(x))'= f'(x)+ g'(x).

4. 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}
$
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