0
$\begingroup$

So I have a partial sum:

$$\sum_{k = 1}^{L}\bigg(\frac{k}{M}\bigg)^{\gamma} \qquad \gamma\in \mathbb{R}, M \in \mathbb{N}$$

As I send $M \rightarrow \infty$ I would like to see how it sends to zero, if at at all. I am dubious to use the integral test, because that seems to apply to series.

  • 0
    Take ${1\over M ^\gamma}$ out of the sum. – 2017-02-05
  • 0
    It's not really a series since $L$ is constant (at least seems to be). Also, if $\gamma<0$ then this sum goes to infinity when $M\to\infty$ and not to zero. – 2017-02-05

1 Answers 1

1

Since your sum is finite, you can distribute the limit like so

$$\lim_{M \to \infty} \sum_{k=1}^L \left( \frac{k}{M} \right)^\gamma = \sum_{k=1}^L \lim_{M \to \infty} \left( \frac{k}{M} \right)^\gamma = \sum_{k=1}^L k^\gamma \lim_{M \to \infty} M^{-\gamma}$$

and the limit is $0$ in the case of $\gamma > 0$, $L$ when $\gamma = 0$, and $\infty$ when $\gamma < 0$.