I would like to find a closed formula for this equation:
$$ \sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y} $$
Both the denominator and also the exponent is changing in each step. How is it possible to simplify it?
If the closed form does not exist for some reason, then I would like to find an approximation.
I have some experience with the sum of arithmetic and geometric progression, but this is something different, and I don't know, how to apply my previous knowledge here.
In my application the value of $n$ is usually between 50 and 1000, therefore $lim_{n \to \infty}$ may give a wrong result. $x$ is in the range $(-10, 10)$ where $x \neq 0$ and $y$ is in the range $(0, \frac{1}{10})$ where $y>0$
One thing is for sure: the terms in the sum converge to a constant, namely $$ \lim_{i\to\infty} \left(\frac{i}{x + i}\right)^{i y} = \frac{1}{\lim_{i\to\infty} \left[\left(1+\frac{x}{i}\right)^i\right]^y} = \frac{1}{[e^x]^y} = e^{-xy} $$ Convergence of the terms may be very slow, depending on $x$. Anyway, it means that the sum itself does not converge to anything: it is continuously increasing.