# Converting an Unconventional Riemann Sum to an Indefinite Integral

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!

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