I am trying to convert the following Riemann Sum into an indefinite integral:

$$\lim_{n \to \infty} \Bigl(nx_{n+1}^m +\sum_{k=1}^n -x_k^m \Bigr)$$

where $m$ is an arbitrary constant. I am wondering if this sum could be converted to an integral in the first place, given that pesky $nx_{n+1}^m$ term before the summation.

I have tried to look at other Riemann Sum-Integral conversions - trapezoidal approximation, center of mass integrals - but none of them seem to help in converting this particular limit. If anyone could help me out here, it would be much appreciated. Thanks!

  • $\begingroup$ This equality might be of help $nx_{n+1}^m-\sum_{k=1}^nx_k^m=\sum_{k=1}^n(n-k+1)(x_{k+1}^m-x_k^m)$. $\endgroup$ – John_Wick Dec 20 '18 at 19:45
  • 1
    $\begingroup$ @John_Wick: I get $\sum_{k=1}^nk\!\left(x_{k+1}^m-x_k^m\right)$ $\endgroup$ – robjohn Dec 20 '18 at 20:27

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