A power series doesn't converge for every x — only for x-values close enough to its
center. This tool shows you that interval, how to find it with the ratio test, and the endpoint
check everyone forgets.
A power series is a polynomial that never stops: Σ aₙ(x−c)ⁿ.
Plug in an x close to the center c and it adds up to a number; plug in an x too far away and it blows up.
The set of x-values where it converges is an interval centered at c — and finding it is a two-part job:
the radius (how far out it works) and the endpoints (which, if any, of the two edges to include).
See the interval
Pick a power series. The number line shows its center, the
converging region, and what happens at each endpoint. Notice
that the two ends don't always behave the same way.
converges herediverges here● filled = endpoint included
Choose a power series
These six cover every case. Messier exam-style series are in the practice section below.
How to find it — the method
Every interval-of-convergence problem is the same three moves. Here they are applied
to the series you've selected above, so the steps track your choice.
Practice — work a full problem
This is the real exam skill: grind the ratio-test algebra yourself, type the radius,
then reason through the endpoints. Pick a difficulty and work all the way to the interval.