sum of_n=1^infinity (1)/(n^3 + 9n + 9)

Compare with sum of 1/n^3; Converges

sum of_n=1^infinity (6n^2 + 7)/(4n^3 + 9)

Compare with sum of 1/n; lim_n to infinity a_n/b_n = 3/2; Diverges

sum of_n=1^infinity (4)/(n× 2^n)

Compare with sum of 4/2^n; Converges

sum of_n=1^infinity (5n + 7)/(3n^2 + 7)

Compare with sum of 1/n; lim_n to infinity a_n/b_n = 5/3; Diverges

sum of_n=3^infinity (6ln n)/(n)

Compare with sum of 6/n; Diverges

sum of_n=1^infinity (8n - 2)/(7n^3 + 9n + 8)

Compare with sum of 1/n^2; lim_n to infinity a_n/b_n = 8/7; Converges

sum of_n=1^infinity (2^n)/(3^n + 3)

Compare with sum of 2^n/3^n; Converges

sum of_n=1^infinity (n^2 + 1)/(3n^4 + 7)

Compare with sum of 1/n^2; lim_n to infinity a_n/b_n = 1/3; Converges

sum of_n=1^infinity (n + 7)/(n^2)

Compare with sum of 1/n; Diverges

sum of_n=1^infinity (2n + 4)/(4n^3 + 5n + 5)

Compare with sum of 1/n^2; lim_n to infinity a_n/b_n = 1/2; Converges

sum of_n=1^infinity (4)/(n^2 + n + 9)

Compare with sum of 4/n^2; Converges

sum of_n=1^infinity (n + 5)/(4n^2 + 2)

Compare with sum of 1/n; lim_n to infinity a_n/b_n = 1/4; Diverges

sum of_n=3^infinity 6ln nsqrt(n)

Compare with sum of 6sqrt(n); Diverges

sum of_n=1^infinity (4n^2 + 7)/(2n^4 + 5)

Compare with sum of 1/n^2; lim_n to infinity a_n/b_n = 2; Converges

sum of_n=1^infinity (4)/(n× 5^n)

Compare with sum of 4/5^n; Converges

sum of_n=1^infinity (4n^2 + 7)/(3n^3 + 3)

Compare with sum of 1/n; lim_n to infinity a_n/b_n = 4/3; Diverges

sum of_n=1^infinity (3^n)/(5^n + 5)

Compare with sum of 3^n/5^n; Converges

sum of_n=1^infinity (8n + 3)/(3n^3 + n + 8)

Compare with sum of 1/n^2; lim_n to infinity a_n/b_n = 8/3; Converges

sum of_n=1^infinity (1)/(n^2 + n + 6)

Compare with sum of 1/n^2; Converges

sum of_n=1^infinity (2n + 5)/(5n^2 + 5)

Compare with sum of 1/n; lim_n to infinity a_n/b_n = 2/5; Diverges

sum of_n=1^infinity (n + 6)/(n^2 + 7)

Compare with sum of 1/n; lim_n to infinity a_n/b_n = 1; Diverges

Worksheets · Calculus

Comparison tests worksheet

Most series behave like a simpler one: drop the small terms and what is left is a p-series or a geometric series. The direct test needs an inequality that holds term by term; the limit test only needs the ratio of the terms to settle on a positive number.

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.



Sigma Prep · sigmaprep.io/worksheets Name Comparison Tests Date Period Name a series to compare with and decide whether each series converges. $\sum_{n = 1}^{\infty} \frac{1}{n^{3} + 9 n + 9}$ $\sum_{n = 1}^{\infty} \frac{6 n^{2} + 7}{4 n^{3} + 9}$ $\sum_{n = 1}^{\infty} \frac{4}{n \cdot 2^{n}}$ $\sum_{n = 1}^{\infty} \frac{5 n + 7}{3 n^{2} + 7}$ $\sum_{n = 3}^{\infty} \frac{6 \ln n}{n}$ $\sum_{n = 1}^{\infty} \frac{8 n - 2}{7 n^{3} + 9 n + 8}$ $\sum_{n = 1}^{\infty} \frac{2^{n}}{3^{n} + 3}$ $\sum_{n = 1}^{\infty} \frac{n^{2} + 1}{3 n^{4} + 7}$ $\sum_{n = 1}^{\infty} \frac{n + 7}{n^{2}}$ $\sum_{n = 1}^{\infty} \frac{2 n + 4}{4 n^{3} + 5 n + 5}$ $\sum_{n = 1}^{\infty} \frac{4}{n^{2} + n + 9}$ $\sum_{n = 1}^{\infty} \frac{n + 5}{4 n^{2} + 2}$ $\sum_{n = 3}^{\infty} \frac{6 \ln n}{\sqrt{n}}$ $\sum_{n = 1}^{\infty} \frac{4 n^{2} + 7}{2 n^{4} + 5}$ $\sum_{n = 1}^{\infty} \frac{4}{n \cdot 5^{n}}$ $\sum_{n = 1}^{\infty} \frac{4 n^{2} + 7}{3 n^{3} + 3}$ $\sum_{n = 1}^{\infty} \frac{3^{n}}{5^{n} + 5}$ $\sum_{n = 1}^{\infty} \frac{8 n + 3}{3 n^{3} + n + 8}$ $\sum_{n = 1}^{\infty} \frac{1}{n^{2} + n + 6}$ $\sum_{n = 1}^{\infty} \frac{2 n + 5}{5 n^{2} + 5}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Comparison Tests

Date Period

Name a series to compare with and decide whether each series converges.

$\sum_{n = 1}^{\infty} \frac{1}{n^{3} + 9 n + 9}$
$\sum_{n = 1}^{\infty} \frac{6 n^{2} + 7}{4 n^{3} + 9}$
$\sum_{n = 1}^{\infty} \frac{4}{n \cdot 2^{n}}$
$\sum_{n = 1}^{\infty} \frac{5 n + 7}{3 n^{2} + 7}$
$\sum_{n = 3}^{\infty} \frac{6 \ln n}{n}$
$\sum_{n = 1}^{\infty} \frac{8 n - 2}{7 n^{3} + 9 n + 8}$
$\sum_{n = 1}^{\infty} \frac{2^{n}}{3^{n} + 3}$
$\sum_{n = 1}^{\infty} \frac{n^{2} + 1}{3 n^{4} + 7}$
$\sum_{n = 1}^{\infty} \frac{n + 7}{n^{2}}$
$\sum_{n = 1}^{\infty} \frac{2 n + 4}{4 n^{3} + 5 n + 5}$
$\sum_{n = 1}^{\infty} \frac{4}{n^{2} + n + 9}$
$\sum_{n = 1}^{\infty} \frac{n + 5}{4 n^{2} + 2}$
$\sum_{n = 3}^{\infty} \frac{6 \ln n}{\sqrt{n}}$
$\sum_{n = 1}^{\infty} \frac{4 n^{2} + 7}{2 n^{4} + 5}$
$\sum_{n = 1}^{\infty} \frac{4}{n \cdot 5^{n}}$
$\sum_{n = 1}^{\infty} \frac{4 n^{2} + 7}{3 n^{3} + 3}$
$\sum_{n = 1}^{\infty} \frac{3^{n}}{5^{n} + 5}$
$\sum_{n = 1}^{\infty} \frac{8 n + 3}{3 n^{3} + n + 8}$
$\sum_{n = 1}^{\infty} \frac{1}{n^{2} + n + 6}$
$\sum_{n = 1}^{\infty} \frac{2 n + 5}{5 n^{2} + 5}$



Sigma Prep · sigmaprep.io/worksheets Name Comparison TestsAnswer keyVersion 1 Date Period Name a series to compare with and decide whether each series converges. $\sum_{n = 1}^{\infty} \frac{1}{n^{3} + 9 n + 9}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{3}} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{6 n^{2} + 7}{4 n^{3} + 9}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{3}{2} \\ Diverges \end{array}$ $\sum_{n = 1}^{\infty} \frac{4}{n \cdot 2^{n}}$ $\begin{array}{l} Compare with \sum \frac{4}{2^{n}} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{5 n + 7}{3 n^{2} + 7}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{5}{3} \\ Diverges \end{array}$ $\sum_{n = 3}^{\infty} \frac{6 \ln n}{n}$ $\begin{array}{l} Compare with \sum \frac{6}{n} \\ Diverges \end{array}$ $\sum_{n = 1}^{\infty} \frac{8 n - 2}{7 n^{3} + 9 n + 8}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{8}{7} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{2^{n}}{3^{n} + 3}$ $\begin{array}{l} Compare with \sum \frac{2^{n}}{3^{n}} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{n^{2} + 1}{3 n^{4} + 7}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{1}{3} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{n + 7}{n^{2}}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ Diverges \end{array}$ $\sum_{n = 1}^{\infty} \frac{2 n + 4}{4 n^{3} + 5 n + 5}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{1}{2} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{4}{n^{2} + n + 9}$ $\begin{array}{l} Compare with \sum \frac{4}{n^{2}} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{n + 5}{4 n^{2} + 2}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{1}{4} \\ Diverges \end{array}$ $\sum_{n = 3}^{\infty} \frac{6 \ln n}{\sqrt{n}}$ $\begin{array}{l} Compare with \sum \frac{6}{\sqrt{n}} \\ Diverges \end{array}$ $\sum_{n = 1}^{\infty} \frac{4 n^{2} + 7}{2 n^{4} + 5}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 2 \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{4}{n \cdot 5^{n}}$ $\begin{array}{l} Compare with \sum \frac{4}{5^{n}} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{4 n^{2} + 7}{3 n^{3} + 3}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{4}{3} \\ Diverges \end{array}$ $\sum_{n = 1}^{\infty} \frac{3^{n}}{5^{n} + 5}$ $\begin{array}{l} Compare with \sum \frac{3^{n}}{5^{n}} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{8 n + 3}{3 n^{3} + n + 8}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{8}{3} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{1}{n^{2} + n + 6}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ Converges \end{array}$ $\sum_{n = 1}^{\infty} \frac{2 n + 5}{5 n^{2} + 5}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{2}{5} \\ Diverges \end{array}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Comparison TestsAnswer keyVersion 1

Date Period

Name a series to compare with and decide whether each series converges.

$\sum_{n = 1}^{\infty} \frac{1}{n^{3} + 9 n + 9}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{3}} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{6 n^{2} + 7}{4 n^{3} + 9}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{3}{2} \\ Diverges \end{array}$
$\sum_{n = 1}^{\infty} \frac{4}{n \cdot 2^{n}}$ $\begin{array}{l} Compare with \sum \frac{4}{2^{n}} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{5 n + 7}{3 n^{2} + 7}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{5}{3} \\ Diverges \end{array}$
$\sum_{n = 3}^{\infty} \frac{6 \ln n}{n}$ $\begin{array}{l} Compare with \sum \frac{6}{n} \\ Diverges \end{array}$
$\sum_{n = 1}^{\infty} \frac{8 n - 2}{7 n^{3} + 9 n + 8}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{8}{7} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{2^{n}}{3^{n} + 3}$ $\begin{array}{l} Compare with \sum \frac{2^{n}}{3^{n}} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{n^{2} + 1}{3 n^{4} + 7}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{1}{3} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{n + 7}{n^{2}}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ Diverges \end{array}$
$\sum_{n = 1}^{\infty} \frac{2 n + 4}{4 n^{3} + 5 n + 5}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{1}{2} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{4}{n^{2} + n + 9}$ $\begin{array}{l} Compare with \sum \frac{4}{n^{2}} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{n + 5}{4 n^{2} + 2}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{1}{4} \\ Diverges \end{array}$
$\sum_{n = 3}^{\infty} \frac{6 \ln n}{\sqrt{n}}$ $\begin{array}{l} Compare with \sum \frac{6}{\sqrt{n}} \\ Diverges \end{array}$
$\sum_{n = 1}^{\infty} \frac{4 n^{2} + 7}{2 n^{4} + 5}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 2 \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{4}{n \cdot 5^{n}}$ $\begin{array}{l} Compare with \sum \frac{4}{5^{n}} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{4 n^{2} + 7}{3 n^{3} + 3}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{4}{3} \\ Diverges \end{array}$
$\sum_{n = 1}^{\infty} \frac{3^{n}}{5^{n} + 5}$ $\begin{array}{l} Compare with \sum \frac{3^{n}}{5^{n}} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{8 n + 3}{3 n^{3} + n + 8}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{8}{3} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{1}{n^{2} + n + 6}$ $\begin{array}{l} Compare with \sum \frac{1}{n^{2}} \\ Converges \end{array}$
$\sum_{n = 1}^{\infty} \frac{2 n + 5}{5 n^{2} + 5}$ $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \frac{2}{5} \\ Diverges \end{array}$

How to do these

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

For large n the terms behave like 2n/n^3, so compare with the sum of 1/n^2.
Divided, the terms give (2n^3 + n^2)/(n^3 + 4), which approaches 2.
The limit is positive and finite, and the p-series with p = 2 converges, so this one does too.

The answer is Compare with $\sum \frac{1}{n^{2}}$ ; $\lim \frac{a_{n}}{b_{n}} = 2$ ; converges.

Where students go wrong

Using the direct test with the inequality pointing the wrong way. Being smaller than a divergent series, or bigger than a convergent one, proves nothing.

Also called direct comparison test, limit comparison test, comparison test for series or basic comparison test.

What you can put on this worksheet

Direct comparison: $\sum_{n = 1}^{\infty} \frac{n + 7}{n^{2}}$ → $\begin{array}{l} Compare with \sum \frac{1}{n} \\ Diverges \end{array}$
Limit comparison: $\sum_{n = 1}^{\infty} \frac{n + 6}{n^{2} + 7}$ → $\begin{array}{l} Compare with \sum \frac{1}{n} \\ \lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 1 \\ Diverges \end{array}$

Questions about these worksheets

Yes. Every comparison tests 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 comparison tests 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: direct comparison and limit comparison. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the direct test with the inequality pointing the wrong way. Being smaller than a divergent series, or bigger than a convergent one, proves nothing.

Comparison tests worksheet

Select your difficulty

Comparison Tests

Comparison TestsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these comparison tests worksheets free?+

Do the comparison tests worksheets come with an answer key?+

Are these comparison tests worksheets printable?+

What grade are these comparison tests worksheets for?+

Can I get a different version of the same comparison tests worksheet?+

Is this a free alternative to Kuta Software for comparison tests?+

Can I choose how hard the comparison tests questions are?+

What kinds of question are on a comparison tests worksheet?+

What do students get wrong with comparison tests?+