Determine the radius of convergence and test convergence
Determine the radius of convergence and test convergence at the endpoints for $\displaystyle\sum_{n=1}^{\infty}\frac{x^n}{a^n+b^n}$, $a,b>0$.
Explanation of Method
For large $n$, the larger of $a,b$ dominates the denominator, so the radius is governed entirely by $M=\max(a,b)$. At $x=\pm M$, the dominant-term ratio $M^n/(a^n+b^n)$ tends to a nonzero constant ($1$ if $a\ne b$, $1/2$ if $a=b$), so both endpoints diverge by the Divergence Test.
- Apply the ratio test to find $R=\max(a,b)$.
- Test $x=M$: the terms tend to a nonzero constant.
- Test $x=-M$: the terms oscillate but never approach $0$.
Step 1 · Radius of convergence
Let $M=\max(a,b)$. Dividing numerator and denominator by $M^{n+1}$ in the ratio test: $$ \lim_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right| = \lim_{n\to\infty}\frac{a^n+b^n}{a^{n+1}+b^{n+1}} = \frac1M $$ (the term with base $M$ dominates as $n\to\infty$, regardless of whether $a=b$ or $a\ne b$). So $R=M=\max(a,b)$.
Step 2 · Endpoint $x=M$
If $a>b$ (so $M=a$): $$ \sum_{n=1}^{\infty}\frac{a^n}{a^n+b^n} = \sum_{n=1}^{\infty}\frac1{1+(b/a)^n} \longrightarrow 1 \ne0 \text{ (term-wise)}, $$ since $(b/a)^n\to0$. If $a=b$: every term is $\dfrac{a^n}{2a^n}=\dfrac12$, a nonzero constant. Either way, the terms do not tend to $0$, so the series diverges by the Divergence Test.
Step 3 · Endpoint $x=-M$
The terms are $(-1)^n$ times the (nonzero-limiting) magnitudes from Step 2, so they oscillate without ever approaching $0$. The series diverges here as well.
Summary Table
| Property | Value |
|---|---|
| Radius of convergence | $R=\max(a,b)$ |
| At $x=\pm\max(a,b)$ | Diverges (terms don't $\to0$) |
| Interval of convergence | $(-\max(a,b),\,\max(a,b))$ |
Additional Notes
- This series generalizes the pure geometric series $\sum(x/a)^n$ (the special case $b\to0$ or, more precisely, whichever of $a,b$ is dominant behaves like a single exponential): the smaller parameter only ever perturbs the constant that the terms converge to at the boundary, never the radius itself.
- The special case $a=b$ collapses the series to exactly $\frac12\sum(x/a)^n$, a scaled geometric series, whose divergence at $x=\pm a$ (where the ratio has magnitude $1$) is the most elementary instance of this problem's general pattern.
- Like Problem 233, this is another example where the Divergence Test alone — rather than any comparison or alternating-series argument — is exactly the right tool to dispatch both endpoints simultaneously.