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I am trying to use induction to prove, for every positive integer $k$, that

$$\sum_{j=1}^{n} {j^{k}} = \frac{n^{k+1}}{k+1} +\frac{n^k}{2} + P_{k-1}(n)$$

where $P_{k-1}$ is a polynomial of degree at most $k-1$.

In order to do the inductive step, I assume that the statement is true for $k$ and use that to prove the statement for $k+1$.

In order to do this I would need to express $\sum_{j=1}^{n} {j^{k+1}}$ in terms of $\sum_{j=1}^{n} {j^{k}}$. How would I do this?

AbsoluteZero2A03
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    If you do it with "strong induction" instead, then you can do this, except that you'll need a bit of extra work to compute the coefficient on $n^k$. http://math.stackexchange.com/a/1194116/83396 – Ian Jan 19 '16 at 18:43
  • Consider finding the $n^{k+1}$ term first by $\sum j^k$ from $\sum j^{k+1}$ – Shrey Aryan Jan 19 '16 at 18:49
  • This question seems related to some extent. (Namely it is a question about inductive proof concerning this sum. You might have a look at the inductive proof given there whether it helps in deriving stronger result you ask for.) – Martin Sleziak Jan 20 '16 at 07:39

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