Convergence of sequences worksheet

A sequence converges when its terms settle on one number as n grows. For a fraction of polynomials, compare the leading terms; for anything else, the limit rules for functions still apply.

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Sigma Prep · sigmaprep.io/worksheets Name Convergence of Sequences Date Period Decide whether each sequence converges. If it does, give its limit. Does the sequence converge? If so, to what? $a_{n} = \frac{6 n^{2} - 4}{6 n + 6}$ Does the sequence converge? If so, to what? $a_{n} = \cos (n π)$ Does the sequence converge? If so, to what? $a_{n} = \frac{- n^{3} - 5}{3 n^{2} + 2}$ Does the sequence converge? If so, to what? $a_{n} = \frac{\ln n}{n}$ Does the sequence converge? If so, to what? $a_{n} = \frac{- 2 n^{2} - 1}{n^{3} + 6}$ Does the sequence converge? If so, to what? $a_{n} = \frac{3 n}{n + 1}$ Does the sequence converge? If so, to what? $a_{n} = \frac{5 n + 4}{2 n^{2} + 5}$ Does the sequence converge? If so, to what? $a_{n} = \frac{6 n}{n + 1}$ Does the sequence converge? If so, to what? $a_{n} = \frac{- 2 n - 5}{3 n^{3} + 2}$ Does the sequence converge? If so, to what? $a_{n} = \frac{n}{2^{n}}$ Does the sequence converge? If so, to what? $a_{n} = \frac{2 n + 4}{6 n^{2} + 4}$ Does the sequence converge? If so, to what? $a_{n} = \frac{\sin n}{n}$ Does the sequence converge? If so, to what? $a_{n} = \frac{6 n^{3} + 8}{6 n^{3} + 2}$ Does the sequence converge? If so, to what? $a_{n} = {(1 + \frac{6}{n})}^{n}$ Does the sequence converge? If so, to what? $a_{n} = \frac{6 n - 6}{5 n^{3} + 7}$ Does the sequence converge? If so, to what? $a_{n} = {(\frac{3}{2})}^{n}$ Does the sequence converge? If so, to what? $a_{n} = \frac{- n^{3} - 3}{3 n + 6}$ Does the sequence converge? If so, to what? $a_{n} = \sqrt{n^{2} + 4 n} - n$ Does the sequence converge? If so, to what? $a_{n} = \frac{2 n - 4}{n^{3} + 9}$ Does the sequence converge? If so, to what? $a_{n} = \frac{4^{n}}{n!}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Convergence of Sequences

Date Period

Decide whether each sequence converges. If it does, give its limit.

Does the sequence converge? If so, to what? $a_{n} = \frac{6 n^{2} - 4}{6 n + 6}$
Does the sequence converge? If so, to what? $a_{n} = \cos (n π)$
Does the sequence converge? If so, to what? $a_{n} = \frac{- n^{3} - 5}{3 n^{2} + 2}$
Does the sequence converge? If so, to what? $a_{n} = \frac{\ln n}{n}$
Does the sequence converge? If so, to what? $a_{n} = \frac{- 2 n^{2} - 1}{n^{3} + 6}$
Does the sequence converge? If so, to what? $a_{n} = \frac{3 n}{n + 1}$
Does the sequence converge? If so, to what? $a_{n} = \frac{5 n + 4}{2 n^{2} + 5}$
Does the sequence converge? If so, to what? $a_{n} = \frac{6 n}{n + 1}$
Does the sequence converge? If so, to what? $a_{n} = \frac{- 2 n - 5}{3 n^{3} + 2}$
Does the sequence converge? If so, to what? $a_{n} = \frac{n}{2^{n}}$
Does the sequence converge? If so, to what? $a_{n} = \frac{2 n + 4}{6 n^{2} + 4}$
Does the sequence converge? If so, to what? $a_{n} = \frac{\sin n}{n}$
Does the sequence converge? If so, to what? $a_{n} = \frac{6 n^{3} + 8}{6 n^{3} + 2}$
Does the sequence converge? If so, to what? $a_{n} = {(1 + \frac{6}{n})}^{n}$
Does the sequence converge? If so, to what? $a_{n} = \frac{6 n - 6}{5 n^{3} + 7}$
Does the sequence converge? If so, to what? $a_{n} = {(\frac{3}{2})}^{n}$
Does the sequence converge? If so, to what? $a_{n} = \frac{- n^{3} - 3}{3 n + 6}$
Does the sequence converge? If so, to what? $a_{n} = \sqrt{n^{2} + 4 n} - n$
Does the sequence converge? If so, to what? $a_{n} = \frac{2 n - 4}{n^{3} + 9}$
Does the sequence converge? If so, to what? $a_{n} = \frac{4^{n}}{n!}$



Sigma Prep · sigmaprep.io/worksheets Name Convergence of SequencesAnswer keyVersion 1 Date Period Decide whether each sequence converges. If it does, give its limit. Does the sequence converge? If so, to what? $a_{n} = \frac{6 n^{2} - 4}{6 n + 6}$ Diverges Does the sequence converge? If so, to what? $a_{n} = \cos (n π)$ Diverges Does the sequence converge? If so, to what? $a_{n} = \frac{- n^{3} - 5}{3 n^{2} + 2}$ Diverges Does the sequence converge? If so, to what? $a_{n} = \frac{\ln n}{n}$ Converges to $0$ Does the sequence converge? If so, to what? $a_{n} = \frac{- 2 n^{2} - 1}{n^{3} + 6}$ Converges to $0$ Does the sequence converge? If so, to what? $a_{n} = \frac{3 n}{n + 1}$ Converges to $3$ Does the sequence converge? If so, to what? $a_{n} = \frac{5 n + 4}{2 n^{2} + 5}$ Converges to $0$ Does the sequence converge? If so, to what? $a_{n} = \frac{6 n}{n + 1}$ Converges to $6$ Does the sequence converge? If so, to what? $a_{n} = \frac{- 2 n - 5}{3 n^{3} + 2}$ Converges to $0$ Does the sequence converge? If so, to what? $a_{n} = \frac{n}{2^{n}}$ Converges to $0$ Does the sequence converge? If so, to what? $a_{n} = \frac{2 n + 4}{6 n^{2} + 4}$ Converges to $0$ Does the sequence converge? If so, to what? $a_{n} = \frac{\sin n}{n}$ Converges to $0$ Does the sequence converge? If so, to what? $a_{n} = \frac{6 n^{3} + 8}{6 n^{3} + 2}$ Converges to $1$ Does the sequence converge? If so, to what? $a_{n} = {(1 + \frac{6}{n})}^{n}$ Converges to $e^{6}$ Does the sequence converge? If so, to what? $a_{n} = \frac{6 n - 6}{5 n^{3} + 7}$ Converges to $0$ Does the sequence converge? If so, to what? $a_{n} = {(\frac{3}{2})}^{n}$ Diverges Does the sequence converge? If so, to what? $a_{n} = \frac{- n^{3} - 3}{3 n + 6}$ Diverges Does the sequence converge? If so, to what? $a_{n} = \sqrt{n^{2} + 4 n} - n$ Converges to $2$ Does the sequence converge? If so, to what? $a_{n} = \frac{2 n - 4}{n^{3} + 9}$ Converges to $0$ Does the sequence converge? If so, to what? $a_{n} = \frac{4^{n}}{n!}$ Converges to $0$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Convergence of SequencesAnswer keyVersion 1

Date Period

Decide whether each sequence converges. If it does, give its limit.

Does the sequence converge? If so, to what? $a_{n} = \frac{6 n^{2} - 4}{6 n + 6}$ Diverges
Does the sequence converge? If so, to what? $a_{n} = \cos (n π)$ Diverges
Does the sequence converge? If so, to what? $a_{n} = \frac{- n^{3} - 5}{3 n^{2} + 2}$ Diverges
Does the sequence converge? If so, to what? $a_{n} = \frac{\ln n}{n}$ Converges to $0$
Does the sequence converge? If so, to what? $a_{n} = \frac{- 2 n^{2} - 1}{n^{3} + 6}$ Converges to $0$
Does the sequence converge? If so, to what? $a_{n} = \frac{3 n}{n + 1}$ Converges to $3$
Does the sequence converge? If so, to what? $a_{n} = \frac{5 n + 4}{2 n^{2} + 5}$ Converges to $0$
Does the sequence converge? If so, to what? $a_{n} = \frac{6 n}{n + 1}$ Converges to $6$
Does the sequence converge? If so, to what? $a_{n} = \frac{- 2 n - 5}{3 n^{3} + 2}$ Converges to $0$
Does the sequence converge? If so, to what? $a_{n} = \frac{n}{2^{n}}$ Converges to $0$
Does the sequence converge? If so, to what? $a_{n} = \frac{2 n + 4}{6 n^{2} + 4}$ Converges to $0$
Does the sequence converge? If so, to what? $a_{n} = \frac{\sin n}{n}$ Converges to $0$
Does the sequence converge? If so, to what? $a_{n} = \frac{6 n^{3} + 8}{6 n^{3} + 2}$ Converges to $1$
Does the sequence converge? If so, to what? $a_{n} = {(1 + \frac{6}{n})}^{n}$ Converges to $e^{6}$
Does the sequence converge? If so, to what? $a_{n} = \frac{6 n - 6}{5 n^{3} + 7}$ Converges to $0$
Does the sequence converge? If so, to what? $a_{n} = {(\frac{3}{2})}^{n}$ Diverges
Does the sequence converge? If so, to what? $a_{n} = \frac{- n^{3} - 3}{3 n + 6}$ Diverges
Does the sequence converge? If so, to what? $a_{n} = \sqrt{n^{2} + 4 n} - n$ Converges to $2$
Does the sequence converge? If so, to what? $a_{n} = \frac{2 n - 4}{n^{3} + 9}$ Converges to $0$
Does the sequence converge? If so, to what? $a_{n} = \frac{4^{n}}{n!}$ Converges to $0$

How to do these

$a_{n} = \frac{3 n^{2} + 1}{n^{2} + 4}$

Top and bottom are both degree 2.
Divide every term by n².
What is left tends to the ratio of the leading coefficients.

The answer is Converges to $3$ .

Where students go wrong

Confusing the sequence with the series. The terms can settle on 3 while their sum runs off to infinity; this page asks only about the terms.

Also called limit of a sequence or convergent and divergent sequences.

What you can put on this worksheet

Rational sequences: Does the sequence converge? If so, to what? $a_{n} = \frac{n^{2} + 1}{3 n^{2} + 3}$ → Converges to $\frac{1}{3}$
Other sequences: Does the sequence converge? If so, to what? $a_{n} = \sqrt{n^{2} + 4 n} - n$ → Converges to $2$

Questions about these worksheets

Yes. Every convergence of sequences 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 convergence of sequences 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: rational sequences and other sequences. Tick as many as you want and set how many of each, or let it spread them evenly.

Confusing the sequence with the series. The terms can settle on 3 while their sum runs off to infinity; this page asks only about the terms.

Convergence of sequences worksheet

Select your difficulty

Convergence of Sequences

Convergence of SequencesAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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