How do I find the limit of the following sequence:
$$\lim_{n\to\infty}\sum_{k=1}^n\frac{k^r}{n^{r+1}}.$$
Asked
Active
Viewed 111 times
2
-
It's a limit of a Riemann sum. – user296113 Dec 10 '16 at 18:08
1 Answers
2
Hint. One may recognize a Riemann sum, by writing, $$ \sum_{k=1}^n\frac{k^r}{n^{r+1}}=\frac{1}{n}\cdot\sum_{k=1}^n\left(\frac{k}{n}\right)^{r}. $$
Olivier Oloa
- 122,789
-
Hope you know this result: http://math.stackexchange.com/questions/1987358/calculate-an-integral-with-riemann-sum – Olivier Oloa Dec 10 '16 at 18:45
-
-
Yes there is. We have $\sum_{k=1}^n k^r= \frac{(n+1)^{r+1}}{r+1} + \cdots$ from which you may deduce the sought limit. Have a look here:https://en.wikipedia.org/wiki/Summation#Some_summations_of_polynomial_expressions or here: https://en.wikipedia.org/wiki/Faulhaber%27s_formula – Olivier Oloa Dec 10 '16 at 20:47