Finding Polynomial Equations for $1^4 + 2^4 + 3^4 + \ldots n^4$ Find a polynomial expression for
$$ 1^4 + 2^4 + 3^4 + ... + n^4 $$
I know you have to use the big theorem but I can't figure out how you would start to compute the differences. Suggestions?

If you know that $f(n) = a_5n^5 + \cdots + a_0$, you can calculate the coefficients using limits:
$$
a_5 = \lim_{n\to\infty}\frac{f(n)}{n^5} =
\lim_{n\to\infty}\frac{1^4+\cdots+n^4}{n^5} =
\lim_{n\to\infty}\frac{(n+1)^4}{(n+1)^5-n^5} = \cdots =
\frac15.
$$
(Cesàro-Stolz used in the third $=$).
Continue with
$$a_4 = \lim_{n\to\infty}\frac{f(n)-a_5n^5}{n^4} = \cdots$$
$$\cdots$$
