Region of uniform convergence

Question

I have to find the region of uniform convergence of

$$e^{x} +2^p\,e^{-2x} +3^p\,e^{−3x} +4^p\,e^{-4x} +\cdots $$

Should I start by saying that the expression is less or larger than a value?

Answer 1

Hint: $\displaystyle\frac{(n+1)^pe^{-(n+1)x}}{n^pe^{-nx}}=\left(1+\frac1n\right)^pe^{-x}$.

Answer 2

From $x\leq 0$ we have $\sum_{k=1}^\infty k^p e^{-kx}\geq \sum_{k=1}^\infty k^p=+\infty$ (assuming $p\geq-1$; for a general $p \in \mathbb{R}$ and $x>0$ you can show that $a_k$ does not tend to 0). In case $x>\epsilon>0$ we can apply the M-test with $M_{k,\epsilon}=k^p e^{-k \epsilon}$, and $\sum_{k=1}^\infty M_{k,\epsilon}<\infty$ via ratio test.

So the series converge uniformly in $I_\epsilon=(\epsilon,+\infty)$ for any $\epsilon >0$.

Edit: Note that in case $ p<-1$ we can set $\epsilon=0 $ and obtaining uniform convergence in $[0,+\infty)$ using $M_{0,k}$.