Interval of Convergence Calculator

A power series centred at $c$ has the form

$\sum_{n=0}^{\infty} a_n (x - c)^n$

and the set of $x$ for which it converges is always one of three shapes: the single point $x = c$, an interval centred at $c$, or the whole real line. The half-width of that interval is the radius of convergence $R$:

$|x - c| < R \Rightarrow \text{converges absolutely}, \qquad |x - c| > R \Rightarrow \text{diverges}$

The interval of convergence is that open interval together with whichever endpoints happen to converge. At $|x - c| = R$ the ratio test is silent, so each endpoint has to be tested separately as an ordinary numerical series — and the two ends can genuinely disagree.

$R = 0$ means convergence at the centre only; $R = \infty$ means everywhere.

The procedure

1. Apply the ratio test to the absolute value of the general term:

$L = \lim_{n \to \infty} \left|\frac{a_{n+1}(x-c)^{n+1}}{a_n(x-c)^n}\right| = |x - c| \cdot \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$

2. Impose $L < 1$ and solve for $|x - c|$. That inequality reads $|x - c| < R$, which hands you the radius directly.
3. Write the open interval $(c - R,\ c + R)$.
4. Test each endpoint by substituting it back into the original series and applying a numerical test — $p$-series, alternating series, or the nth-term test.
5. Assemble the interval, using a square bracket for each endpoint that converged.

Shortcuts

When $a_n$ contains an $n$-th power, the root test $R = 1/\lim \sqrt[n]{|a_n|}$ is faster. A factorial in the denominator drives $L$ to $0$ and gives $R = \infty$; a factorial in the numerator drives it to $\infty$ and gives $R = 0$.

A useful sanity check: the series always converges at its own centre, where every term after the first is zero.

• Stopping at the radius: "$R = 1$" is not an interval. The endpoints are where most of the marks are.
• Assuming both endpoints behave alike: $\sum \frac{(x-2)^n}{n}$ converges at one end and diverges at the other.
• Using the ratio test at an endpoint: there $L = 1$ and the test is inconclusive by construction. Substitute the value and pick a different test.
• Forgetting to solve for $x$: $|x - c| < R$ still has to be unwrapped into $c - R < x < c + R$.
• Mishandling a coefficient inside the power: for $\sum 3^n x^n$ the condition is $3|x| < 1$, so $R = \frac{1}{3}$, not $3$.
• Ignoring the centre: the interval is centred at $c$, not at $0$, whenever the series is written in $(x - c)$.