How to find the interval of convergence of a power series quickly

Expert intro: When the terms of a power series grow factorially, the Ratio Test is usually the fastest route. Here the series is so explosive that the convergence picture becomes very short.

Detailed walkthrough

Consider the power series with general term $a_k=k!x^k.$ To study convergence, apply the Ratio Test:

$$r=\lim_{k\to\infty}\frac{|a_{k+1}|}{|a_k|} =\lim_{k\to\infty}\frac{|(k+1)!x^{k+1}|}{|k!x^k|}.$$

Simplify:

$$r=\lim_{k\to\infty}|x|(k+1).$$

Now split into cases.

If $x\neq 0$

Then $|x|>0$, so

$$r=\infty,$$

which is greater than 1. The series diverges.

If $x=0$

Then every term is zero for $k\ge1$, so the series converges trivially.

So the interval of convergence is just

$$\{0\}.$$

The radius of convergence is

$$R=0.$$

General method to remember

For series with factorials, powers, exponentials, or products, the Ratio Test is often the first thing to try. If the ratio tends to a number less than 1, you get convergence; if it is greater than 1, divergence; if it equals 1, you need another test.

💡 Pitfall guide

A frequent mistake is to think a power series always has some positive radius of convergence. Not true. Factorials in the numerator can destroy convergence completely, leaving only the center point. Another small trap is forgetting to test the center separately when the ratio limit is not informative.

🔄 Real-world variant

If the term were $\frac{x^k}{k!}$ instead of $k!x^k$, the same Ratio Test would give a limit of $0$ for every real $x$, so the radius of convergence would be infinite. Swapping factorials from numerator to denominator completely changes the behavior.

🔍 Related terms

Ratio Test, radius of convergence, power series