Using Cauchy's root test, examine the convergence of: sum
Using Cauchy's root test, examine the convergence of: $$ \sum_{n=1}^{\infty}\left(\frac{x}{a_n}\right)^n, $$ where $\lim_{n\to\infty} a_n = a$, $x>0$, and $a_n>0$.
Explanation of Method
This generalizes Problem 72 (where $a_n=n\to\infty$ forced automatic convergence) by letting $a_n$ tend to a finite limit $a$. The root test simplifies exactly to $\sqrt[n]{b_n}=x/a_n\to x/a$, so the outcome now depends on comparing $x$ to $a$, with a genuine threshold at $x=a$ and a separate degenerate case when $a=0$.
- Apply the root test to get $\sqrt[n]{b_n}=x/a_n$.
- Evaluate $\rho=x/a$ for $a>0$, and handle $a=0$ separately.
- Analyze the borderline case $x=a$.
Step 1 · Apply the root test
With $b_n=\left(\dfrac x{a_n}\right)^n$:
$$ \sqrt[n]{b_n} = \frac{x}{a_n}. $$
Step 2 · Evaluate the limit
Case $a>0$: since $a_n\to a$, $\rho=\lim_{n\to\infty}\dfrac x{a_n}=\dfrac xa$.
Case $a=0$: then $\rho=\lim_{n\to\infty}\dfrac x{a_n}=+\infty>1$, so the series diverges for every $x>0$.
Step 3 · Apply the criterion and analyze $x=a$
For $a>0$: converges if $x<a$ ($\rho<1$); diverges if $x>a$ ($\rho>1$); inconclusive if $x=a$ ($\rho=1$). At $x=a$, write $a_n=a(1+\varepsilon_n)$ with $\varepsilon_n\to0$, so $b_n=(1+\varepsilon_n)^{-n}$; the outcome then depends on the rate of $\varepsilon_n\to0$ — e.g. $a_n=a+c/n$ gives $b_n\to e^{-c}$ (diverges), while $a_n=a-c/n$ gives $b_n\to e^{c}$ (diverges), so this borderline case generally requires finer analysis of how $a_n$ approaches $a$.
Summary Table
| Case | $\rho$ | Conclusion |
|---|---|---|
| $a>0,\;x<a$ | $x/a<1$ | Converges |
| $a>0,\;x>a$ | $x/a>1$ | Diverges |
| $a>0,\;x=a$ | $1$ | Inconclusive (needs finer analysis) |
| $a=0$ | $+\infty$ | Diverges for all $x>0$ |
Additional Notes
- Setting $a_n\equiv n$ recovers Problem 72 as the limiting degenerate sub-case $a=+\infty$ (formally outside this problem's $a>0$ or $a=0$ split), which is consistent with $\rho=x/a\to0$ as $a\to\infty$.
- The borderline analysis at $x=a$ shows that Cauchy's root test alone cannot resolve every case — the specific rate at which $a_n\to a$ becomes essential, echoing the general lesson from Problem 80 that $\rho=1$ is truly inconclusive without extra information.
- This problem is a template for a whole family of "perturbed geometric-like" series: whenever the base of an $n$-th power tends to a finite nonzero limit, the root test reduces the whole series to a simple comparison of that limit against $1$ (after rescaling by $x$).
Result
For $a>0$, the series converges iff $x<a$; for $a=0$, it diverges for all $x>0$: