Topic 10.13 Radius and Interval of Convergence of Power Series: find the radius and interval of convergence of a power series using the ratio test and checking the endpoints (BC).

What this topic is asking

The College Board (Topic 10.13, BC only) studies power series $\sum c_n(x - a)^n$, which are "infinite polynomials" in $x$. A power series converges for some values of $x$ and diverges for others; this topic finds the interval of $x$-values where it converges, using the ratio test and a careful check of the two endpoints.

The method

Why the ratio test gives the radius

The ratio test is the natural tool because a power series has the $n$-th-power structure the ratio test handles best. Forming $\left|\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}\right|$, the powers combine to a single factor $|x - a|$, leaving $|x - a|\cdot\lim\left|\frac{c_{n+1}}{c_n}\right|$. Setting this $< 1$ for absolute convergence isolates $|x - a|$ below a threshold $R$, the radius. So the ratio test does not just decide one series; for a power series it produces an inequality in $x$ that defines the convergence set. The endpoints, where the ratio-test limit equals exactly $1$, are precisely the inconclusive case, which is why they always need separate handling.

A worked radius and interval

The endpoints are where the work is

Finding the radius is usually quick; the endpoints are where most of the marks and most of the errors live. At each endpoint the ratio-test limit is exactly $1$, so the ratio test says nothing, and you must substitute the endpoint $x$-value back into the series and apply a different test. Common outcomes: a $p$-series, the harmonic series (diverges), an alternating series (often converges by the alternating series test), or a geometric series. Each endpoint can behave differently, as the worked example shows, so test them independently. The final interval includes an endpoint with a square bracket only if the series converges there. Skipping endpoint analysis, or assuming both endpoints behave the same, is the classic loss of credit.

The three possible radii

Every power series falls into exactly one of three cases for $R$. If the ratio-test limit is $0$ for all $x$ (as for $\sum\frac{x^n}{n!}$, since the factorial dominates), the series converges everywhere and $R = \infty$; there are no endpoints to test. If the limit is $\infty$ for all $x\neq a$ (as for $\sum n!\,x^n$), the series converges only at the center $x = a$ and $R = 0$. Otherwise $0 < R < \infty$ and you get a finite interval requiring endpoint analysis. Recognizing which case you are in from the ratio-test limit tells you immediately whether endpoint work is even needed: only the finite-$R$ case has endpoints.