Q:

Find the value of $c$ such that $\sum\limits_{n=1}^{\infty} e^{nc} = 10$

Find the value of $c$ such that $$\sum_{n=1}^{\infty} e^{nc} = 10$$

I don't even know how to start this. 
This is a geometric series so inorder for it to converge to any number it has to be $|e^{nc}| < 1$
my attempt:
$$|e^{nc}| < 1$$
$$|nc| < \ln 1$$

A:

By the geometric series formula, that sum is $\frac{e^{c}}{1-e^c}$. Notice that the series only converges for $c<0$, so our answer is only valid if it gives that $c$ is less than $0$. Setting this equal to $10$ and doing a bit of algebra gives us $10e^c+e^c=10\Rightarrow e^c=\frac{10}{11}$. Thus the answer is $c=\ln(10)-\ln(11)<0$

