The ratio test gives the radius: the power series converges inside it and diverges outside. At the two endpoints the ratio test gives 1 and decides nothing, so each one is a separate series to test.
Select your difficulty
Pick more than one for a sheet that mixes them.
The preview can look cramped on a phone. The PDF and printed sheet come out normal.
The ratio of terms approaches |x - 2|/3, so the series converges when |x - 2| < 3: R = 3.
At x = 5 the series is the harmonic series, which diverges.
At x = -1 it is the alternating harmonic series, which converges.
The answer is R=3; interval: [−1,5).
Where students go wrong
Leaving the endpoints out, or testing only one of them. Each endpoint is its own series, and the two often behave differently.
Also called interval of convergence, radius of convergence or power series convergence.
What you can put on this worksheet
The radius of convergence
Radius: n=1∑∞n(x−1)n→ R=1
The interval, endpoints checked
Radius and interval: n=1∑∞n(x−1)n→ R=1Interval: [0,2)
Questions about these worksheets
Yes. Every radius and interval of convergence sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the radius and interval of convergence questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 2: the radius of convergence and the interval, endpoints checked. Tick as many as you want and set how many of each, or let it spread them evenly.
Leaving the endpoints out, or testing only one of them. Each endpoint is its own series, and the two often behave differently.