Prove that:

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

and/or that

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

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- Feb 5th 2010, 10:05 AMBraveHeartPower rule.
Prove that:

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

and/or that

$\displaystyle \int\left(\sum_{k=0}^{n}a_{k}x^{k}\right){dx} = \sum_{k=0}^{n}\frac{a_{k}x^{k+1}}{k+1}+c$ - Feb 5th 2010, 02:03 PMTKHunny
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. - Feb 6th 2010, 04:21 AMHallsofIvy
No, completely right. Induction on n and the "sum rule": (f(x)+ g(x))'= f'(x)+ g'(x).

- Feb 10th 2010, 09:20 PMBraveHeart
I'm not sure how to proceed with induction.

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

$.