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