Ratio and Root Test

Question 1

Ratio Test: Determine the convergence of $\sum_{n=1}^{\infty} \frac{n!}{n^n}$.

Answer 1

Converges. The limit $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$ simplifies to $\lim (1 + 1/n)^{-n} = 1/e$, which is $< 1$.

Question 2

Root Test: Determine the convergence of $\sum_{n=1}^{\infty} (\frac{n-1}{n})^{n^2}$.

Answer 2

Converges. The limit $\lim_{n \to \infty} \sqrt[n]{|a_n|}$ is $\lim (1 - 1/n)^n = 1/e$, which is $< 1$.

Question 3

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{(2n)!}{(n!)^2}$.

Answer 3

Diverges. The ratio $\frac{a_{n+1}}{a_n}$ simplifies to $\frac{(2n+2)(2n+1)}{(n+1)^2}$, which has a limit of $4 > 1$.

Question 4

Root Test: Determine the convergence of $\sum_{n=1}^{\infty} \frac{n^2}{2^n}$.

Answer 4

Converges. The limit $\lim_{n \to \infty} \frac{(\sqrt[n]{n})^2}{2} = 1/2$, which is $< 1$.

Question 5

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{3^n n!}{n^n}$.

Answer 5

Diverges. The limit of the ratio is $3/e \approx 3/2.718$, which is $> 1$.

Question 6

Root Test: Analyze $\sum_{n=1}^{\infty} (\frac{2n+3}{3n+2})^n$.

Answer 6

Converges. The limit $\lim_{n \to \infty} \frac{2n+3}{3n+2} = 2/3$, which is $< 1$.

Question 7

Conceptual: If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = 1$, can the Root Test provide a conclusive result?

Answer 7

No. If the Ratio Test is inconclusive ($L=1$) for a standard algebraic/power series, the Root Test will also be inconclusive ($L=1$).

Question 8

Ratio Test: Determine the convergence of $\sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!}$.

Answer 8

Converges. The ratio simplifies to $\frac{(n+1)^2}{(2n+2)(2n+1)}$, which has a limit of $1/4 < 1$.

Question 9

Root Test: Analyze $\sum_{n=1}^{\infty} (\ln(e + \frac{1}{n}))^n$.

Answer 9

Diverges. The limit $\lim_{n \to \infty} \ln(e + 1/n) = \ln(e) = 1$, making the test inconclusive. (However, the terms approach $1$, so it diverges by the nth-term test).

Question 10

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{100^n}{n!}$.

Answer 10

Converges. The ratio is $\frac{100}{n+1}$, which approaches $0 < 1$.

Question 11

Root Test: Determine the convergence of $\sum_{n=1}^{\infty} (1 + \frac{1}{n})^{-n^2}$.

Answer 11

Converges. The limit is $\lim (1 + 1/n)^{-n} = 1/e < 1$.

Question 12

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{n!}{(2n+1)!}$.

Answer 12

Converges. The ratio is $\frac{n+1}{(2n+3)(2n+2)}$, which approaches $0 < 1$.

Question 13

Root Test: Analyze $\sum_{n=1}^{\infty} (\frac{n}{n+1})^n$.

Answer 13

Inconclusive. The limit $\lim \frac{n}{n+1}$ is $1$. (Note: The series diverges by the nth-term test because the terms approach $1/e$).

Question 14

Conceptual: Which test is typically more efficient for series involving factorials?

Answer 14

The Ratio Test, because factorials cancel out cleanly in the fraction $\frac{a_{n+1}}{a_n}$.

Question 15

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{(n+1)!}{e^{n^2}}$.

Answer 15

Converges. The ratio $\frac{n+2}{e^{2n+1}}$ approaches $0$ because exponential growth in the denominator dominates the linear numerator.

Question 16

Root Test: Determine the convergence of $\sum_{n=1}^{\infty} (\frac{\arctan n}{n})^n$.

Answer 16

Converges. The limit is $\lim \frac{\arctan n}{n} = 0 < 1$.

Question 17

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{4^n (n!)^2}{(2n)!}$.

Answer 17

Inconclusive. The limit of the ratio is $\lim \frac{4(n+1)^2}{(2n+2)(2n+1)} = 1$.

Question 18

Root Test: Analyze $\sum_{n=1}^{\infty} \frac{2^{n^2}}{n!}$.

Answer 18

Diverges. The limit $\lim \frac{2^n}{\sqrt[n]{n!}}$ approaches $\infty$ because $2^n$ grows much faster than the average value of the factors in the factorial.

Question 19

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \dots (2n-1)}{2 \cdot 4 \cdot 6 \dots (2n)}$.

Answer 19

Inconclusive. The ratio is $\frac{2n+1}{2n+2}$, which approaches $1$.

Question 20

Root Test: Determine the convergence of $\sum_{n=1}^{\infty} \frac{n^k}{k^n}$ for $k > 1$.

Answer 20

Converges. The limit is $\lim \frac{(\sqrt[n]{n})^k}{k} = 1/k$, which is $< 1$ since $k > 1$.

Question 21

Conceptual: If a series $\sum a_n$ converges by the Ratio Test, does $\sum |a_n|$ also converge?

Answer 21

Yes. The Ratio Test naturally tests for absolute convergence because it uses the absolute value of the terms.

Question 22

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{(n!)^2 2^n}{(2n)!}$.

Answer 22

Converges. The limit of the ratio is $2 \cdot (1/4) = 1/2 < 1$.

Question 23

Root Test: Analyze $\sum_{n=1}^{\infty} (\frac{n}{2n+1})^n$.

Answer 23

Converges. The limit is $1/2 < 1$.

Question 24

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{n^3}{3^n}$.

Answer 24

Converges. The ratio $\frac{(n+1)^3}{3n^3}$ approaches $1/3 < 1$.

Question 25

Root Test: Analyze $\sum_{n=1}^{\infty} (\sqrt[n]{n} - 1)^n$.

Answer 25

Converges. The limit is $\lim (\sqrt[n]{n} - 1) = 1 - 1 = 0 < 1$.

Question 26

Ratio Test: Analyze $\sum_{n=1}^{\infty} \frac{2^n n!}{(n+2)!}$.

Answer 26

Diverges. The ratio $\frac{2(n+1)}{n+3}$ approaches $2 > 1$.

Question 27

Root Test: Analyze $\sum_{n=1}^{\infty} \frac{e^n}{n^n}$.

Answer 27

Converges. The limit $\lim e/n$ is $0 < 1$.

Question 28

Ratio Test: Find the radius of convergence $R$ for $\sum_{n=1}^{\infty} \frac{x^n}{n!}$.

Answer 28

$R = \infty$. The Ratio Test shows the limit is $0$ for any $x$, so it converges for all real numbers.

Question 29

Root Test: Analyze $\sum_{n=1}^{\infty} (1 - \frac{2}{n})^{n^2}$.

Answer 29

Converges. The limit $\lim (1 - 2/n)^n$ is $e^{-2} = 1/e^2 < 1$.

Question 30

Conceptual: If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L > 1$, what can you conclude about $\lim_{n \to \infty} a_n$?

Answer 30

The limit $\lim_{n \to \infty} a_n \ne 0$ (in fact, the terms grow infinitely in magnitude), so the series diverges by the nth-term test.