sum or polynomial of a polynomial?

Question

I am trying to answer the following math problem but I barely understand the question.

If $P(i)$ is a polynomial of degree $d$ in $i$ then $\sum\limits_{i=0}^n p(i)$ is a polynomial of $d + 1$ in $n$. Imagine that $P(i) = 2i^3 + 4i^2 + 2$ what is $\sum\limits_{i=0}^n p(i)$

Could somebody please give me some pointers on what they want to know? Or what "tools" I can use to fix this problem? Do I need to use Newtons Binomial theorem?

Thanks in advance

Answer 1

$$ \sum_{i=0}^n (2i^3 + 4i^2 + 2) = 2\left( \sum_{i=0}^n i^3 \right) + 4\left( \sum_{i=0}^n i^2 \right) + 2\left( \sum_{i=0}^n 2 \right). $$ Therefore, it is enough to deal with each of $\displaystyle\sum_{i=0}^n i^k$ for $k=0,1,2,3,\ldots$. For that we need the Bernoulli polynomials $B_n$, characterized by $$ \int_x^{x+1} B_n(u)\,du = x^n. $$ We have $$ \sum_{i=0}^n i^k = \frac{B_{k+1}(i+1)-B_{k+1}(0)}{k+1}. $$