11.6E: Exercises for Section 11.6

In exercises 1-8, use the ratio test to determine whether each series \(\displaystyle \sum^∞_{n=1}a_n\) converges or diverges. State if the ratio test is inconclusive.

1) \(\displaystyle \sum_{n=1}^∞\frac{1}{n!}\)

Answer

\(\displaystyle R=\lim_{n\to \infty}\frac{a_{n+1}}{a_n}=0<1.\) Series is convergent by the Ratio Test.

2) \(\displaystyle \sum_{n=1}^∞\frac{10^n}{n!}\)

Answer

Series is convergent by the Ratio Test; R=0.

3) \(\displaystyle \sum_{n=1}^∞\frac{n^2}{2^n}\)

Answer

\(\displaystyle R=\lim_{n\to \infty} \frac{a_{n+1}}{a_n}=\lim_{n\to \infty}\frac{1}{2}\left(\frac{n+1}{n}\right)^2=\frac{1}{2}<1.\) Series is convergent by the Ratio Test.

4) \(\displaystyle \sum_{n=1}^∞\frac{(n!)^3}{(3n)!}\)

Answer

\(\displaystyle R=\lim_{n\to \infty} \frac{a_{n+1}}{a_n}=\frac{1}{27}<1.\) Series is convergent by the Ratio Test.

5) \(\displaystyle \sum_{n=1}^∞\frac{2^{3n}(n!)^3}{(3n)!}\)

Answer

Series is convergent by the Ratio Test.

6) \(\displaystyle \sum_{n=1}^∞\frac{(2n)!}{n^{2n}}\)

Answer

\(\displaystyle \lim_{n\to \infty} \frac{a_{n+1}}{a_n}=\frac{4}{e^2}<1.\) Converges by the Ratio Test.

7) \(\displaystyle \sum_{n=1}^∞\frac{(2n)!}{(2n)^n}\)

Answer

Series is divergent by the Ratio Test.

8) \(\displaystyle \sum_{n=1}^∞\frac{(2^nn!)^2}{(2n)^{2n}}\)

Answer

\(\displaystyle \lim_{n\to \infty} \frac{a_n}{a_{n+1}}=\frac{1}{e^2}<1.\) Converges by the Ratio Test.

Use the root test to determine whether \(\displaystyle \sum^∞_{n=1}a_n\) converges, where \(a_n\) is as follows.

9) \(\displaystyle a_k=\left(\frac{k−1}{2k+3}\right)^k\)

Answer

Series is convergent by the root test.

10) \(\displaystyle a_k=\left(\frac{2k^2−1}{k^2+3}\right)^k\)

Answer

Series is divergent by the root test.

Use either the ratio test or the root test as appropriate to determine whether the series \(\displaystyle \sum_{k=1}^∞a_k\) with given terms \(a_k\) converges, or state if the test is inconclusive.

11) \(\displaystyle a_k=\frac{k!}{1⋅3⋅5⋯(2k−1)}\)

Answer

Series is convergent.

12) \(\displaystyle a_k=\frac{2⋅4⋅6⋯2k}{(2k)!}\)

Answer

\(\displaystyle \lim_{k\to \infty} \frac{a_{k+1}}{a_k}= \lim_{k\to \infty} \frac{1}{2k+1}=0.\) Converges by the Ratio Test.

13) \(\displaystyle a_k=\frac{1⋅4⋅7⋯(3k−2)}{3^kk!}\)

Answer

Test is inconclusive.

Use the ratio test to determine whether \(\displaystyle \sum_{n=1}^∞a_n\) converges, or state if the ratio test is inconclusive.

14) \(\displaystyle \sum_{n=1}^∞\frac{3^{n^2}}{2^{n^3}}\)

Answer

\(\displaystyle \lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right|= \lim_{n\to \infty} \frac{3^{2n+1}}{2^{3n^2+3n+1}}=0.\) Converges by the Ratio Test.

Use an appropriate test to determine whether the series converges.

15) \(\displaystyle \sum_{n=1}^∞\frac{n+1}{n^3+n^2+n+1}\)

Answer

Series is convergent.

16) \(\displaystyle \sum_{n=1}^∞\frac{(−1)^{n+1}(n+1)}{n^3+3n^2+3n+1}\)

Answer

Converges absolutely by limit comparison with \(p\)−series, \(p=2.\) Series is absolutely convergent.

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