Ratio and root tests worksheet

Both tests measure how fast the terms shrink compared with a geometric series. Below 1 the series converges, above 1 it diverges, and at exactly 1 the test cannot tell. Factorials call for the ratio test; a whole term raised to the n calls for the root test.

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Sigma Prep · sigmaprep.io/worksheets Name Ratio and Root Tests Date Period Use the test named to find L, then say what the test shows: converges, diverges, or inconclusive. Ratio test: $\sum_{n = 1}^{\infty} \frac{{(n!)}^{2}}{(2 n)!}$ Root test: $\sum_{n = 1}^{\infty} {(1 + \frac{4}{n})}^{n^{2}}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{4 n!}{(2 n)!}$ Root test: $\sum_{n = 1}^{\infty} {(\frac{4 n^{2} + 3}{2 n^{2} + 5})}^{n}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{n!}{5^{n}}$ Root test: $\sum_{n = 1}^{\infty} \frac{{(1 + \frac{1}{n})}^{n^{2}}}{3^{n}}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{n!}{n^{n}}$ Root test: $\sum_{n = 1}^{\infty} \frac{n^{4}}{6^{n}}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{n^{n}}{n!}$ Root test: $\sum_{n = 1}^{\infty} {(\frac{3 n + 9}{7 n + 4})}^{n}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{(2 n)!}{{(n!)}^{2}}$ Root test: $\sum_{n = 1}^{\infty} \frac{n}{8^{n}}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{3^{n}}{(n + 3)!}$ Root test: $\sum_{n = 1}^{\infty} {(\frac{4 n^{2} + 9}{5 n^{2} + 9})}^{n}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{n!}{4^{n}}$ Root test: $\sum_{n = 1}^{\infty} {(\frac{n + 8}{5 n + 4})}^{n}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{3 {(n!)}^{2}}{(2 n)!}$ Root test: $\sum_{n = 1}^{\infty} \frac{{(1 + \frac{1}{n})}^{n^{2}}}{2^{n}}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{5 (2 n)!}{{(n!)}^{2}}$ Root test: $\sum_{n = 1}^{\infty} {(1 + \frac{3}{n})}^{n^{2}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Ratio and Root Tests

Date Period

Use the test named to find L, then say what the test shows: converges, diverges, or inconclusive.

Ratio test:
$\sum_{n = 1}^{\infty} \frac{{(n!)}^{2}}{(2 n)!}$
Root test:
$\sum_{n = 1}^{\infty} {(1 + \frac{4}{n})}^{n^{2}}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{4 n!}{(2 n)!}$
Root test:
$\sum_{n = 1}^{\infty} {(\frac{4 n^{2} + 3}{2 n^{2} + 5})}^{n}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{n!}{5^{n}}$
Root test:
$\sum_{n = 1}^{\infty} \frac{{(1 + \frac{1}{n})}^{n^{2}}}{3^{n}}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{n!}{n^{n}}$
Root test:
$\sum_{n = 1}^{\infty} \frac{n^{4}}{6^{n}}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{n^{n}}{n!}$
Root test:
$\sum_{n = 1}^{\infty} {(\frac{3 n + 9}{7 n + 4})}^{n}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{(2 n)!}{{(n!)}^{2}}$
Root test:
$\sum_{n = 1}^{\infty} \frac{n}{8^{n}}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{3^{n}}{(n + 3)!}$
Root test:
$\sum_{n = 1}^{\infty} {(\frac{4 n^{2} + 9}{5 n^{2} + 9})}^{n}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{n!}{4^{n}}$
Root test:
$\sum_{n = 1}^{\infty} {(\frac{n + 8}{5 n + 4})}^{n}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{3 {(n!)}^{2}}{(2 n)!}$
Root test:
$\sum_{n = 1}^{\infty} \frac{{(1 + \frac{1}{n})}^{n^{2}}}{2^{n}}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{5 (2 n)!}{{(n!)}^{2}}$
Root test:
$\sum_{n = 1}^{\infty} {(1 + \frac{3}{n})}^{n^{2}}$



Sigma Prep · sigmaprep.io/worksheets Name Ratio and Root TestsAnswer keyVersion 1 Date Period Use the test named to find L, then say what the test shows: converges, diverges, or inconclusive. Ratio test: $\sum_{n = 1}^{\infty} \frac{{(n!)}^{2}}{(2 n)!}$ $\begin{array}{l} L = \frac{1}{4} \\ Converges \end{array}$ Root test: $\sum_{n = 1}^{\infty} {(1 + \frac{4}{n})}^{n^{2}}$ $\begin{array}{l} L = e^{4} \\ Diverges \end{array}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{4 n!}{(2 n)!}$ $\begin{array}{l} L = 0 \\ Converges \end{array}$ Root test: $\sum_{n = 1}^{\infty} {(\frac{4 n^{2} + 3}{2 n^{2} + 5})}^{n}$ $\begin{array}{l} L = 2 \\ Diverges \end{array}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{n!}{5^{n}}$ $\begin{array}{l} L = \infty \\ Diverges \end{array}$ Root test: $\sum_{n = 1}^{\infty} \frac{{(1 + \frac{1}{n})}^{n^{2}}}{3^{n}}$ $\begin{array}{l} L = \frac{e}{3} \\ Converges \end{array}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{n!}{n^{n}}$ $\begin{array}{l} L = \frac{1}{e} \\ Converges \end{array}$ Root test: $\sum_{n = 1}^{\infty} \frac{n^{4}}{6^{n}}$ $\begin{array}{l} L = \frac{1}{6} \\ Converges \end{array}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{n^{n}}{n!}$ $\begin{array}{l} L = e \\ Diverges \end{array}$ Root test: $\sum_{n = 1}^{\infty} {(\frac{3 n + 9}{7 n + 4})}^{n}$ $\begin{array}{l} L = \frac{3}{7} \\ Converges \end{array}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{(2 n)!}{{(n!)}^{2}}$ $\begin{array}{l} L = 4 \\ Diverges \end{array}$ Root test: $\sum_{n = 1}^{\infty} \frac{n}{8^{n}}$ $\begin{array}{l} L = \frac{1}{8} \\ Converges \end{array}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{3^{n}}{(n + 3)!}$ $\begin{array}{l} L = 0 \\ Converges \end{array}$ Root test: $\sum_{n = 1}^{\infty} {(\frac{4 n^{2} + 9}{5 n^{2} + 9})}^{n}$ $\begin{array}{l} L = \frac{4}{5} \\ Converges \end{array}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{n!}{4^{n}}$ $\begin{array}{l} L = \infty \\ Diverges \end{array}$ Root test: $\sum_{n = 1}^{\infty} {(\frac{n + 8}{5 n + 4})}^{n}$ $\begin{array}{l} L = \frac{1}{5} \\ Converges \end{array}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{3 {(n!)}^{2}}{(2 n)!}$ $\begin{array}{l} L = \frac{1}{4} \\ Converges \end{array}$ Root test: $\sum_{n = 1}^{\infty} \frac{{(1 + \frac{1}{n})}^{n^{2}}}{2^{n}}$ $\begin{array}{l} L = \frac{e}{2} \\ Diverges \end{array}$ Ratio test: $\sum_{n = 1}^{\infty} \frac{5 (2 n)!}{{(n!)}^{2}}$ $\begin{array}{l} L = 4 \\ Diverges \end{array}$ Root test: $\sum_{n = 1}^{\infty} {(1 + \frac{3}{n})}^{n^{2}}$ $\begin{array}{l} L = e^{3} \\ Diverges \end{array}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Ratio and Root TestsAnswer keyVersion 1

Date Period

Use the test named to find L, then say what the test shows: converges, diverges, or inconclusive.

Ratio test:
$\sum_{n = 1}^{\infty} \frac{{(n!)}^{2}}{(2 n)!}$ $\begin{array}{l} L = \frac{1}{4} \\ Converges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} {(1 + \frac{4}{n})}^{n^{2}}$ $\begin{array}{l} L = e^{4} \\ Diverges \end{array}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{4 n!}{(2 n)!}$ $\begin{array}{l} L = 0 \\ Converges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} {(\frac{4 n^{2} + 3}{2 n^{2} + 5})}^{n}$ $\begin{array}{l} L = 2 \\ Diverges \end{array}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{n!}{5^{n}}$ $\begin{array}{l} L = \infty \\ Diverges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} \frac{{(1 + \frac{1}{n})}^{n^{2}}}{3^{n}}$ $\begin{array}{l} L = \frac{e}{3} \\ Converges \end{array}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{n!}{n^{n}}$ $\begin{array}{l} L = \frac{1}{e} \\ Converges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} \frac{n^{4}}{6^{n}}$ $\begin{array}{l} L = \frac{1}{6} \\ Converges \end{array}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{n^{n}}{n!}$ $\begin{array}{l} L = e \\ Diverges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} {(\frac{3 n + 9}{7 n + 4})}^{n}$ $\begin{array}{l} L = \frac{3}{7} \\ Converges \end{array}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{(2 n)!}{{(n!)}^{2}}$ $\begin{array}{l} L = 4 \\ Diverges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} \frac{n}{8^{n}}$ $\begin{array}{l} L = \frac{1}{8} \\ Converges \end{array}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{3^{n}}{(n + 3)!}$ $\begin{array}{l} L = 0 \\ Converges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} {(\frac{4 n^{2} + 9}{5 n^{2} + 9})}^{n}$ $\begin{array}{l} L = \frac{4}{5} \\ Converges \end{array}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{n!}{4^{n}}$ $\begin{array}{l} L = \infty \\ Diverges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} {(\frac{n + 8}{5 n + 4})}^{n}$ $\begin{array}{l} L = \frac{1}{5} \\ Converges \end{array}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{3 {(n!)}^{2}}{(2 n)!}$ $\begin{array}{l} L = \frac{1}{4} \\ Converges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} \frac{{(1 + \frac{1}{n})}^{n^{2}}}{2^{n}}$ $\begin{array}{l} L = \frac{e}{2} \\ Diverges \end{array}$
Ratio test:
$\sum_{n = 1}^{\infty} \frac{5 (2 n)!}{{(n!)}^{2}}$ $\begin{array}{l} L = 4 \\ Diverges \end{array}$
Root test:
$\sum_{n = 1}^{\infty} {(1 + \frac{3}{n})}^{n^{2}}$ $\begin{array}{l} L = e^{3} \\ Diverges \end{array}$

How to do these

$\sum_{n = 1}^{\infty} \frac{3^{n}}{n!}$

Divide the (n + 1)th term by the nth: 3^(n+1)/(n+1)! times n!/3^n.
Everything cancels to 3/(n + 1), which approaches 0.
L = 0 is less than 1, so the series converges.

The answer is $L = 0$ ; converges.

Where students go wrong

Deciding a series with L = 1. Every p-series has L = 1 under both tests, and some converge while others diverge, so L = 1 means use a different test.

Also called d'Alembert's ratio test, Cauchy's root test, ratio test, root test or nth root test.

What you can put on this worksheet

The ratio test: Ratio test:
$\sum_{n = 1}^{\infty} \frac{6^{n}}{n}$ → $\begin{array}{l} L = 6 \\ Diverges \end{array}$
The root test: Root test:
$\sum_{n = 1}^{\infty} {(\frac{6}{n})}^{n}$ → $\begin{array}{l} L = 0 \\ Converges \end{array}$

Questions about these worksheets

Yes. Every ratio and root 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 ratio and root 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: the ratio test and the root test. Tick as many as you want and set how many of each, or let it spread them evenly.

Deciding a series with L = 1. Every p-series has L = 1 under both tests, and some converge while others diverge, so L = 1 means use a different test.

Ratio and root tests worksheet

Select your difficulty

Ratio and Root Tests

Ratio and Root TestsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

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