Power Series: Radius and Uniform Convergence

A power series $\sum c_n x^{n}$ converges on an interval centered at $0$, as you saw in Calculus 2. The root test with lim sup gives the radius in every case, and the M-test shows that the sum is continuous inside.

Key idea

Cauchy–Hadamard. $\sum c_n x^{n}$ has radius of convergence $R = \frac{1}{\limsup |c_n|^{1/n}}$ (with $\frac{1}{0} = \infty$ and $\frac{1}{\infty} = 0$): it converges absolutely for $|x| < R$ and diverges for $|x| > R$. For every $r < R$ it converges uniformly on $[-r, r]$, so its sum is continuous on $(-R, R)$. The endpoints must be checked separately.

Why it works

Apply the root test to the terms $c_n x^{n}$: $\limsup |c_n x^{n}|^{1/n} = |x|\limsup |c_n|^{1/n} = \frac{|x|}{R}$. This is less than $1$ when $|x| < R$ (convergence) and more than $1$ when $|x| > R$ (divergence). For uniform convergence on $[-r, r]$ with $r < R$: the number $r$ itself is inside the radius, so $\sum |c_n| r^{n}$ converges, and $M_n = |c_n| r^{n}$ bounds $|c_n x^{n}|$ for every $|x| \le r$. The M-test finishes the proof. $\blacksquare$

The lim sup matters for series with gaps or irregular coefficients. In $\sum 4^{n} x^{2n}$ the odd coefficients are $0$, so the ratios $\frac{c_{n+1}}{c_n}$ are undefined half the time; but as a series in $x$, $|c_{2m}|^{1/(2m)} = 4^{1/2} = 2$ and the odd ones are $0$, so $\limsup = 2$ and $R = \frac{1}{2}$.

The differentiated series $\sum n c_n x^{n-1}$ has the same radius, because $n^{1/n} \to 1$ does not change the lim sup. On each $[-r, r]$ with $r < R$ it converges uniformly, so by the derivative theorem of the previous topic, a power series can be differentiated term by term inside its interval, as many times as you like.

Worked examples

Worked example: A series with gaps

Find the radius of convergence of $\sum_{n=0}^{\infty} 9^{n} x^{2n}$.

As a series in $x$: $c_{2n} = 9^{n}$ and the odd coefficients are $0$. So $|c_{2n}|^{1/(2n)} = 3$.
$\limsup |c_k|^{1/k} = 3$, so $R = \frac{1}{3}$.

Answer: $\frac{1}{3}$ (check: it is geometric in $9x^{2}$, which needs $9x^{2} < 1$).

Worked example: Coefficients that alternate rules

Find the radius for $c_n = 2^{n}$ (even $n$), $c_n = 5^{n}$ (odd $n$).

$|c_n|^{1/n}$ is $2$ or $5$; the lim sup is $5$.

Answer: $R = \frac{1}{5}$.

Worked example: The endpoints

Find the interval of convergence of $\sum_{n=1}^{\infty} \frac{(x - 1)^{n}}{n\,3^{n}}$.

$|c_n|^{1/n} = \frac{1}{n^{1/n}\,3} \to \frac{1}{3}$, so $R = 3$ around the center $1$: $(-2, 4)$.
At $x = 4$: $\sum \frac{1}{n}$ diverges. At $x = -2$: $\sum \frac{(-1)^{n}}{n}$ converges.

Answer: $[-2, 4)$

Common mistake

Forcing the ratio test. For series with missing powers or two interleaved patterns, $\frac{c_{n+1}}{c_n}$ has no limit (or is undefined), and "the ratio test fails" is sometimes misread as "the radius does not exist." The root test with lim sup always gives $R$. And the theorem says nothing at $|x| = R$: test the endpoints with other tools.

Power Series: Radius and Uniform Convergence

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