evaluating infinite sum for $e^{x}$

Question

I need some help understanding how to evaluate this: $\sum^{\infty}_{n=1}e^{(x+y)n}$.

I know that as n tends towards infinity, the results gets larger but how do I simplify so that I lose the summation sign?

Answer 1

Since $e^{(x+y)n}=\left(e^{x+y}\right)^n$, your sum is the sum of a geometric progression.

Answer 2

As Jose pointed out you have a geometric series $\sum a^i$. A geometric series will converge to $\frac{a}{ 1-a}$ if $|a|<1$. You can show that this is the requirement for convergence using the D'Alembert ratio test. In your case $a=e^{x+y}$ hence $$|e^{x+y}| < 1 \Leftrightarrow $$ $$e^{x+y} < 1 \Leftrightarrow (e^x >0)$$ $$x+y < log(1) \Leftrightarrow$$ $$ x+y < 0$$