Bernoulli stated sum of series of powers as:
LINK to the image source (Power Sum)
I had a doubt in the given formula in the picture! What if $n < p$ i.e. $1^4 + 2^4 + 3^4$ here $n = 3$ and $p = 4$ so outer summation runs ($i = 1$ to $p$) from $1$ to $4$. And we have to find $(n-i)!$ [factorial]. Won't it lead to factorial of negative number when $i > n$ ?
There is no factorial in the formula, there are binomial coefficients. If you look up the definition carefully, you will note that for nonnegative integers $n,k$ we have
$${n \choose k} := \begin{cases} \frac{n!}{k!(n-k)!}, & k \leq n \\ 0 & else\end{cases}.$$
Thus, you will not run into trouble as there will be no factorials in your case.