Determine the region of absolute and conditional
Determine the region of absolute and conditional convergence of the series: $$ \sum_{n=1}^{\infty} ne^{-n/x}. $$
Explanation of Method
Writing $r=e^{-1/x}$ reduces this to the classical series $\sum nr^n$, which converges precisely for $|r|<1$. Since $e^{-1/x}$ is always positive, the condition $r<1$ simplifies directly to a sign condition on $x$, with no need to separately consider $r<-1$.
- Substitute $r=e^{-1/x}$ and identify $\sum nr^n$.
- Determine when $r<1$ in terms of the sign of $x$.
- Confirm absolute convergence (all terms positive for $x>0$) and give the closed form.
Step 1 · Substitution
With $r=e^{-1/x}>0$, the series is $\sum_{n=1}^{\infty}nr^n$, which converges for $r<1$ (the standard threshold for $\sum nr^n$, here restricted to positive $r$).
Step 2 · Sign condition on $x$
$e^{-1/x}<1\iff-1/x<0\iff1/x>0\iff x>0$. For $x<0$: $-1/x>0$ so $r=e^{-1/x}>1$, and the series diverges (terms grow without bound). At $x=0$: undefined.
Step 3 · Absolute convergence and closed form
For $x>0$ all terms are positive, so convergence is automatically absolute. Using $\sum_{n=1}^{\infty}nr^n=\dfrac r{(1-r)^2}$: $$ \sum_{n=1}^{\infty}ne^{-n/x} = \frac{e^{-1/x}}{(1-e^{-1/x})^2} = \frac{e^{1/x}}{(e^{1/x}-1)^2}. $$
Summary Table
| Region | Behavior |
|---|---|
| $x>0$ | Absolutely convergent, sum $=e^{1/x}/(e^{1/x}-1)^2$ |
| $x\le0$ | Diverges (or undefined at $x=0$) |
Additional Notes
- This is the exponential-substitution analogue of Problem 164's $\sum n/x^n$: both series reduce to the identical $\sum nr^n$ template, but the map $x\mapsto e^{-1/x}$ here restricts the effective ratio $r$ to positive values only, collapsing what would otherwise be a two-sided condition into a simple half-line.
- As $x\to0^+$, $r\to0$ and the series converges extremely rapidly; as $x\to+\infty$, $r\to1^-$ and the sum grows without bound, reflecting the approach to the boundary of convergence.
- Since all terms are positive throughout the domain of convergence, this series — like Problem 169 — has no possibility of conditional convergence.
Result
Converges absolutely for $x>0$; diverges otherwise: