TBIL-Calc Ratio and Root Tests (SQ7)

Skip to main content

Section 8.7 Ratio and Root Tests (SQ7)

Learning Outcomes

🔗
Use the ratio and root tests to determine if a series converges or diverges.

Subsection 8.7.1 Activities

Activity 8.7.1.

Consider the series

\sum_{n = 0}^{\infty} \frac{2^{n}}{3^{n} - 2} .

(a)

Which of these series most closely resembles

\sum_{n = 0}^{\infty} \frac{2^{n}}{3^{n} - 2} ?

🔗
$\sum_{n = 0}^{\infty} \frac{2}{3} .$
🔗
$\sum_{n = 0}^{\infty} \frac{2}{3} n .$
🔗
$\sum_{n = 0}^{\infty} {(\frac{2}{3})}^{n} .$

(b)

Based on your previous choice, do we think this series is more likely to converge or diverge?

(c)

Find

lim_{n \to \infty} \frac{\frac{2^{n + 1}}{3^{n + 1} - 2}}{\frac{2^{n}}{3^{n} - 2}} = lim_{n \to \infty} \frac{2^{n + 1} (3^{n} - 2)}{(3^{n + 1} - 2) 2^{n}} = lim_{n \to \infty} \frac{2 \cdot 2^{n} (3^{n} - 2)}{3 (3^{n} - \frac{2}{3}) 2^{n}} .

🔗
$lim_{n \to \infty} \frac{\frac{2^{n + 1}}{3^{n + 1} - 2}}{\frac{2^{n}}{3^{n} - 2}} = 0 .$
🔗
$lim_{n \to \infty} \frac{\frac{2^{n + 1}}{3^{n + 1} - 2}}{\frac{2^{n}}{3^{n} - 2}} = \frac{2}{3} .$
🔗
$lim_{n \to \infty} \frac{\frac{2^{n + 1}}{3^{n + 1} - 2}}{\frac{2^{n}}{3^{n} - 2}} = 1 .$
🔗
$lim_{n \to \infty} \frac{\frac{2^{n + 1}}{3^{n + 1} - 2}}{\frac{2^{n}}{3^{n} - 2}} = 2 .$
🔗
$lim_{n \to \infty} \frac{\frac{2^{n + 1}}{3^{n + 1} - 2}}{\frac{2^{n}}{3^{n} - 2}} = 3 .$

Activity 8.7.2.

Consider the series

\sum_{n = 0}^{\infty} a_{n} = \sum_{n = 0}^{\infty} \frac{3}{2^{n}} .

(a)

Does

\sum_{n = 0}^{\infty} a_{n} = \sum_{n = 0}^{\infty} \frac{3}{2^{n}}

converge?

(b)

Find

\frac{a_{n + 1}}{a_{n}} .

🔗
$2 .$
🔗
$\frac{1}{2} .$
🔗
$\frac{2^{n}}{2^{n} + 1} .$
🔗
$\frac{9}{2^{2 n + 1}} .$
🔗
$\frac{9}{2^{n + 2}} .$

(c)

Find

lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | .

🔗
$- \infty .$
🔗
$0 .$
🔗
$\frac{1}{2} .$
🔗
$2 .$
🔗
$\infty .$

Activity 8.7.3.

Consider the series

\sum_{n = 1}^{\infty} a_{n} = \sum_{n = 1}^{\infty} \frac{n^{2}}{n + 1} .

(a)

Does

\sum_{n = 1}^{\infty} a_{n} = \sum_{n = 1}^{\infty} \frac{n^{2}}{n + 1}

converge?

(b)

Find

\frac{a_{n + 1}}{a_{n}} .

🔗
$\frac{n + 1}{2} .$
🔗
$\frac{(n^{2} + 1) (n + 1)}{(n + 2) n^{2}} .$
🔗
$\frac{(n + 1)^{2}}{n + 2} .$
🔗
$\frac{1}{2} .$
🔗
$\frac{(n + 1) n^{2}}{n + 2} .$

(c)

Find

lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | .

🔗
$- \infty .$
🔗
$0 .$
🔗
$\frac{1}{2} .$
🔗
$2 .$
🔗
$\infty .$

Activity 8.7.4.

Consider the series

\sum_{n = 1}^{\infty} a_{n} = \sum_{n = 1}^{\infty} \frac{1}{n} .

(a)

Does

\sum_{n = 1}^{\infty} a_{n} = \sum_{n = 1}^{\infty} \frac{1}{n}

converge?

(b)

Find

\frac{a_{n + 1}}{a_{n}} .

(c)

Find

lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | .

Activity 8.7.5.

Consider the series

\sum_{n = 1}^{\infty} a_{n} = \sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{n} .

(a)

Does

\sum_{n = 1}^{\infty} a_{n} = \sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{n}

converge?

(b)

Find

\frac{a_{n + 1}}{a_{n}} .

(c)

Find

lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | .

Fact 8.7.6. The Ratio Test.

Let

\sum a_{n}

be a series and suppose that

lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | = ρ .

Then

🔗
$\sum a_{n}$ converges if $ρ$ is less than 1, and
🔗
$\sum a_{n}$ diverges if $ρ$ is greater than 1.
🔗
If $ρ = 1,$ we cannot determine if $\sum a_{n}$ converges or diverges with this method.

Fact 8.7.7. The Root Test.

Let

N

be an integer and let

\sum a_{n}

be a series with

a_{n} \geq 0

for

n \geq N,

and suppose that

lim_{n \to \infty} \sqrt[n]{| a_{n} |} = ρ .

Then

🔗
$\sum a_{n}$ converges if $ρ$ is less than 1, and
🔗
$\sum a_{n}$ diverges if $ρ$ is greater than 1.
🔗
If $ρ = 1,$ we cannot determine if $\sum a_{n}$ converges or diverges with this method.

Activity 8.7.8.

Consider the series

\sum_{n = 0}^{\infty} \frac{n^{2}}{n!} .

(a)

Which of the following is

a_{n} ?

🔗
$n^{2} .$
🔗
$n! .$
🔗
$\frac{n^{2}}{n!} .$

(b)

Which of the following is

a_{n + 1} ?

🔗
$\frac{n^{2}}{n!} .$
🔗
$(n + 1)^{2} .$
🔗
$(n + 1)! .$
🔗
$\frac{(n + 1)^{2}}{(n + 1)!} .$
🔗
$\frac{n^{2} + 1}{n! + 1} .$

(c)

Which of the following is

| \frac{a_{n + 1}}{a_{n}} | ?

🔗
$\frac{(n + 1)^{2} n^{2}}{(n + 1)! n!} .$
🔗
$\frac{(n + 1)^{2} n!}{(n + 1)! n^{2}} .$
🔗
$\frac{(n + 1)! n!}{(n + 1)^{2} n^{2}} .$
🔗
$\frac{(n + 1)! n^{2}}{(n + 1)^{2} n!} .$

(d)

Using the fact

(n + 1)! = (n + 1) \cdot n!,

simplify

| \frac{a_{n + 1}}{a_{n}} |

as much as possible.

(e)

Find

lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | .

(f)

Does

\sum_{n = 0}^{\infty} \frac{n^{2}}{n!}

converge?

Activity 8.7.9.

(a)

What is

a_{n} ?

(b)

Which of the following is

\sqrt[n]{| a_{n} |} ?

🔗
$\frac{n + 1}{9} .$
🔗
$\frac{n}{9} .$
🔗
$n .$
🔗
$9 .$
🔗
$\frac{1}{9} .$

(c)

Find

lim_{n \to \infty} \sqrt[n]{| a_{n} |} .

(d)

Does

\sum_{n = 1}^{\infty} \frac{n^{n}}{9^{n}}

converge?

Activity 8.7.10.

For each series, use the ratio or root test to determine if the series converges or diverges.

(a)

\sum_{n = 1}^{\infty} {(\frac{1}{1 + n})}^{n}

(b)

\sum_{n = 1}^{\infty} \frac{2^{n}}{n^{n}}

(c)

\sum_{n = 1}^{\infty} \frac{(2 n)!}{(n!) (n!)}

(d)

\sum_{n = 1}^{\infty} \frac{4^{n} (n!) (n!)}{(2 n)!}

Activity 8.7.11.

Consider the series

\sum_{n = 0}^{\infty} \frac{2^{n} + 5}{3^{n}} .

(a)

Use the root test to check for convergence of this series.

(b)

Use the ratio test to check for convergence of this series.

(c)

Use the comparison (or limit comparison) test to check for convergence of this series.

(d)

Find the sum of this series.

Activity 8.7.12.

Consider

\sum_{n = 1}^{\infty} \frac{n}{3^{n}} .

Recall that

\sqrt[n]{\frac{n}{3^{n}}} = {(\frac{n}{3^{n}})}^{1 / n} = \frac{n^{1 / n}}{(3^{n})^{1 / n}} .

(a)

Let

α = lim_{n \to \infty} \ln (n^{1 / n}) = lim_{n \to \infty} \frac{1}{n} \ln (n) .

Find

α .

(b)

Recall that

lim_{n \to \infty} n^{1 / n} = lim_{n \to \infty} e^{\ln (n^{1 / n})} = e^{α} .

Find

lim_{n \to \infty} n^{1 / n} .

(c)

Find

lim_{n \to \infty} \sqrt[n]{\frac{n}{3^{n}}} = lim_{n \to \infty} {(\frac{n}{3^{n}})}^{1 / n} = lim_{n \to \infty} \frac{n^{1 / n}}{(3^{n})^{1 / n}} .

(d)

Does

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

converge?

Activity 8.7.13.

Consider the series

\sum_{n = 0}^{\infty} \frac{n^{2}}{2^{n}} .

(a)

Use the root test to check for convergence of this series.

(b)

Use the ratio test to check for convergence of this series.

(c)

Use the comparison (or limit comparison) test to check for convergence of this series.

Subsection 8.7.2 Videos

Figure 177. Video: Use the ratio and root tests to determine if a series converges or diverges

<Prev ^Top Next>