Nice approximations of sums by integrals.

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Let $f(x):\Bbb Z^+\rightarrow \Bbb R^+$ be a non-monotone function.

If for every $m\in\Bbb N$, $$S(m) =\sum_{n=1}^N\frac{1}{(1+f(n))^m}$$ be sum of interest, then is there a way to approximate this summation using integrals?