How to find the limit of this convergent sum?

2018-03-27 11:25:04

I have a sum $\sum_{i=0}^{\infty} i^{k} D^{i}$, where $k = \frac{1}{2} n$ for some $n \in \mathbb{N}$ and $1>D>0$. The ratio test proves that this sum converges, but I struggle finding a closed form for the limit. Any suggestions how to get there?

The explicit expression for the sum is:

$$

\sum_{i=0}^{\infty} i^{k} D^{i}=\frac{\sum_{i=0}^{k-1}A_i^{k}D^{i+1}}{(1-D)^{k+1}},

$$

where $A_i^k$ are the Eulerian numbers. Can be easily proved by induction.

  • The explicit expression for the sum is:

    $$

    \sum_{i=0}^{\infty} i^{k} D^{i}=\frac{\sum_{i=0}^{k-1}A_i^{k}D^{i+1}}{(1-D)^{k+1}},

    $$

    where $A_i^k$ are the Eulerian numbers. Can be easily proved by induction.

    2018-03-27 14:02:30