Determine the region of absolute and conditional
Determine the region of absolute and conditional convergence of the series: $$ \sum_{n=1}^{\infty} \frac{3^n}{x^n}. $$
Explanation of Method
This is directly a geometric series $\sum(3/x)^n$, so the elementary geometric convergence criterion $|3/x|<1$ applies immediately, without needing the ratio test as a separate step. Since it is purely geometric, there is again no region of conditional convergence: convergence and absolute convergence coincide exactly.
- Recognize the series as geometric with ratio $3/x$.
- Apply the geometric series convergence criterion.
- Check both endpoints and confirm the closed form.
Step 1 · Recognize the geometric series
$\displaystyle\sum_{n=1}^{\infty}\frac{3^n}{x^n}=\sum_{n=1}^{\infty}\left(\frac3x\right)^n$, geometric with ratio $r=3/x$.
Step 2 · Apply the convergence criterion
Converges iff $|r|=|3/x|<1$, i.e. $|x|>3$; diverges for $|x|\le3$.
Step 3 · Check endpoints and closed form
At $x=3$: series becomes $\sum1$, diverges. At $x=-3$: series becomes $\sum(-1)^n$, oscillates, diverges. For $|x|>3$: $$ \sum_{n=1}^{\infty}\left(\frac3x\right)^n = \frac{3/x}{1-3/x} = \frac3{x-3}. $$
Summary Table
| Region | Behavior |
|---|---|
| $|x|>3$ | Absolutely convergent, sum $=3/(x-3)$ |
| $|x|\le3$ | Diverges |
Additional Notes
- This is the simplest possible member of the "$n$-power-of-$x$" family closing out this sequence (Problems 163–165): purely geometric, with no logarithmic (163) or polynomial-weighted (164) complication, and consequently no boundary subtlety whatsoever.
- The threshold $|x|>3$ scales directly with the base of the geometric numerator: replacing $3$ with any constant $c>0$ shifts the threshold to $|x|>c$, by the identical argument.
- Together, Problems 163, 164, and 165 form a deliberate progression in convergence-region subtlety: a half-open interval with genuine conditional convergence at one endpoint (163), a fully closed exclusion with no conditional convergence despite a nontrivial coefficient (164), and a purely geometric case with the simplest possible convergence criterion (165).
Result
Converges absolutely for $|x|>3$, with no conditional convergence: