Evaluate by a Riemann sum: $ \lim_{n \rightarrow \infty}\left(\frac {(n+1)^m+(n+2)^m+\cdots+(n+k)^m}{n^{m-1}}-kn\right)$

HINT:

$$\sum_{r=1}^k(n+r)^m=kn^m+\binom m1n^{m-1}\sum_{r=1}^kr+O(n^{m-2})$$

$$\frac {(n+1)^m+(n+2)^m+\cdots+(n+k)^m}{n^{m-1}}-kn$$ $$=\dfrac{\binom m1n^{m-1}\sum_{r=1}^kr+O(n^{m-2})}{n^{m-1}}$$ $$=\binom m1\sum_{r=1}^kr+O\left(\dfrac1n\right)$$

Tags:

Limits

Real Analysis

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