Approximating the logarithm of sum

Question

I would like to approximate $$ \ln(\sum_{k=0}^n(n-2k)^p) $$ Here $p\geq 2$

Answer 1

Presumably the upper limit should be $n/2$ to avoid powers of negative numbers (particularly problematic if $p$ isn't an odd integer). I would convert to an integral: $\sum_{k=0}^n(n-2k)^p \approx \int_{k=0}^{n/2}(n-2k)^p\;dk$ which yields to the substitution $u=n-2k$.