When is $f(x)=\sum_{n=1}^{\infty}ne^{-nx}$ defined and continuous

Question

So $f(x)=\sum_{n=1}^{\infty}ne^{-nx}$. For which values of $x$ is $f$ defined? What about continuity?

Answer 1

You need your series convergent for $f(x)$ to be defined. And your series is only convergent when $x>0$. You can also actually evaluate the series: \begin{align} f(x)&=\sum_{n=0}^{\infty} ne^{-nx} \\ &=-\sum_{n=0}^{\infty} \frac{\partial}{\partial x} e^{-nx} \\ &= -\frac{\partial}{\partial x}\sum_{n=0}^{\infty} e^{-nx} \\ &= -\frac{\partial }{\partial x} \left(\frac{1}{1-e^{-x}}\right) \\ &=\frac{e^{-x}}{\left(1-e^{-x}\right)^2} \\ &=\frac{e^x}{\left(e^x-1\right)^2} \end{align} Your series is then continuous on $(0,\infty)$.