Determine the region of absolute and conditional
Determine the region of absolute and conditional convergence of the series: $$ \sum_{n=1}^{\infty} \frac1{x^2+n^2}. $$
Explanation of Method
Since $x^2\ge0$, every term satisfies $0<\frac1{x^2+n^2}\le\frac1{n^2}$, and the convergent $p$-series $\sum1/n^2$ immediately gives convergence by direct comparison, for every real $x$ simultaneously. Because all terms are already positive, absolute convergence and ordinary convergence coincide.
- Bound each term above by $1/n^2$.
- Apply the comparison test against the convergent $p$-series $\sum1/n^2$.
- Note that positivity of all terms makes convergence automatically absolute.
Step 1 · Bound the terms
For any real $x$ and $n\ge1$: $\dfrac1{x^2+n^2}\le\dfrac1{n^2}$, since $x^2\ge0$.
Step 2 · Comparison test
$\sum_{n=1}^{\infty}1/n^2$ converges ($p$-series, $p=2>1$), so by direct comparison $\sum_{n=1}^{\infty}1/(x^2+n^2)$ converges for every $x\in\mathbb R$.
Step 3 · Absolute convergence is automatic
Since every term is positive, $\left|\dfrac1{x^2+n^2}\right|=\dfrac1{x^2+n^2}$, so convergence and absolute convergence coincide — the series converges absolutely for all real $x$. At $x=0$: $\sum1/n^2=\pi^2/6$ (the Basel problem); for general $x\ne0$, the closed form is $\dfrac{\pi\coth(\pi|x|)-1/|x|}{2|x|}$ (via the classical partial-fraction expansion of $\coth$).
Summary Table
| Region | Behavior |
|---|---|
| All $x\in\mathbb R$ | Absolutely convergent |
| Special case $x=0$ | Sum $=\pi^2/6$ |
Additional Notes
- This is the simplest possible convergence problem in this batch: no parameter regime causes divergence, since the extra term $x^2$ in the denominator only ever helps (making terms smaller), never hurts, convergence.
- The closed form via $\coth$ connects this series to the classical partial-fraction expansion of the hyperbolic cotangent, a cornerstone identity in the theory of the Weierstrass elliptic functions and theta functions.
- Unlike Problems 163, 166, and 168 (all of which have a genuine boundary of convergence in the parameter), this series' "parameter" $x$ never leaves the region of convergence — a useful contrast illustrating that not every parametrized series problem has an interesting boundary.
Result
The series converges absolutely for every real $x$: