4.6E: Exercises
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Exercise 4.6E.1
Use the ratio test to determine whether ∞∑n=1an converges, where an is given in the following problems. State if the ratio test is inconclusive.
1) an=1/n!
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Solution: an+1/an→0. Converges.
2) an=10n/n!
3) an=n2/2n
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Solution: an+1an=12(n+1n)2→1/2<1. Converges.Add texts here. Do not delete this text first.
4) an=n10/2n
5) ∞∑n=1(n!)3(3n!)
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Solution: an+1an→1/27<1. Converges.
6) ∞∑n=123n(n!)3(3n!)
7) ∞∑n=1(2n)!n2n
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Solution: an+1an→4/e2<1. Converges.
8) ∞∑n=1(2n)!(2n)n
9) ∞∑n=1n!(n/e)n
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Solution: an+1an→1. Ratio test is inconclusive.
10) ∞∑n=1(2n)!(n/e)2n
11) ∞∑n=1(2nn!)2(2n)2n
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Solution: anan+1→1/e2. Converges.
Exercise 4.6E.2
Use the root test to determine whether ∞∑n=1an converges, where an is as follows.
1) ak=(k−12k+3)k
2) ak=(2k2−1k2+3)k
- Answer
Solution: (ak)1/k→2>1. Diverges.
3) an=(lnn)2nnn
4) an=n/2n
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Solution: (an)1/n→1/2<1. Converges
5) an=n/en
6) ak=keek
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Solution: (ak)1/k→1/e<1. Converges.
7) ak=πkkπ
8) an=(1e+1n)n
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Solution: a1/nn=1e+1n→1e<1. Converges.
9) ak=1(1+lnk)k
10) an=(ln(1+lnn))n(lnn)n
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Solution: a1/nn=(ln(1+lnn))(lnn)→0 by L’Hôpital’s rule. Converges.
Exercise 4.6E.3
In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series ∞∑k=1ak with given terms ak converges, or state if the test is inconclusive.
1) ak=k!1⋅3⋅5⋯(2k−1)
2) ak=2⋅4⋅6⋯2k(2k)!
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Solution: ak+1ak=12k+1→0. Converges by ratio test
3) ak=1⋅4⋅7⋯(3k−2)3kk!
4) a)n=(1−1n)n2
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Solution: (an)1/n→1/e. Converges by root test.
5) ak=(1k+1+1k+2+⋯+12k)k (Hint: Compare a1/kk to ∫2kkdtt.)
6) ak=(1k+1+1k+2+⋯+13k)k
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Solution: a1/kk→ln(3)>1. Diverges by root test.
7) an=(n1/n−1)n
Exercise 4.6E.4
Use the ratio test to determine whether ∞∑n=1an converges, or state if the ratio test is inconclusive.
1) ∞∑n=13n22n3
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Solution: an+1an=32n+123n2+3n+1→0. Converge.
2) ∞∑n=12n2nnn!
Exercise 4.6E.5
Use the root and limit comparison tests to determine whether ∞∑n=1an converges.
1) an=1/xnn where xn+1=12xn+1xn,x1=1 (Hint: Find limit of xn.)
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Solution: Converges by root test and limit comparison test since xn→√2.
Exercise 4.6E.6
In the following exercises, use an appropriate test to determine whether the series converges.
1) ∞∑n=1(n+1)n3+n2+n+1
2) ∞∑n=1(−1)n+1(n+1)n3+3n2+3n+1
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Solution: Converges absolutely by limit comparison with p−series, \(\displaystyle p=2.\
3) ∞∑n=1(n+1)2n3+(1.1)n
4) ∞∑n=1(n−1)n(n+1)n
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Solution: lim. Series diverges.
5) \displaystyle a_n=(1+\frac{1}{n^2})^n (Hint: \displaystyle (1+\frac{1}{n^2})^{n^2}≈e.)
6) \displaystyle a_k=1/2^{sin^2k}
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Solution: Terms do not tend to zero: \displaystyle a_k≥1/2, since \(\displaystyle sin^2x≤1.\
7) \displaystyle a_k=2^{−sin(1/k)}
8) \displaystyle a_n=1/(^{n+2}_n) where \displaystyle (^n_k)=\frac{n!}{k!(n−k)!}
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Solution: \displaystyle a_n=\frac{2}{(n+1)(n+2)}, which converges by comparison with p−series for \displaystyle p=2
9) \displaystyle a_k=1/(^{2k}_k)
10) \displaystyle a_k=2^k/(^{3k}_k)
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Solution: \displaystyle a_k=\frac{2^k1⋅2⋯k}{(2k+1)(2k+2)⋯3k}≤(2/3)^k converges by comparison with geometric series
11) \displaystyle a_k=(\frac{k}{k+lnk})^k (Hint: \displaystyle a_k=(1+\frac{lnk}{k})^{−(k/lnk)lnk}≈e^{−lnk}.)
12) \displaystyle a_k=(\frac{k}{k+lnk})^{2k} (Hint: \displaystyle a_k=(1+\frac{lnk}{k})^{−(k/lnk)lnk^2}.)
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Solution: \displaystyle a_k≈e^{−lnk^2}=1/k^2. Series converges by limit comparison with \displaystyle p−series, p=2.
Exercise \PageIndex{7}
The following series converge by the ratio test. Use summation by parts, \displaystyle \sum_{k=1}^na_k(b_{k+1}−b_k)=[a_{n+1}b_{n+1}−a_1b_1]−\sum_{k=1}^nb_{k+1}(a_{k+1}−a_k), to find the sum of the given series.
1) \displaystyle \sum_{k=1}^∞\frac{k}{2^k} (Hint: Take \displaystyle a_k=k and \displaystyle b_k=2^{1−k}.)
2) \displaystyle \sum_{k=1}^∞\frac{k}{c^k}, where \displaystyle c>1 (Hint: Take \displaystyle a_k=k and \displaystyle b_k=c^{1−k}/(c−1).)
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Solution: If \displaystyle b_k=c^{1−k}/(c−1) and \displaystyle a_k=k, then \displaystyle b_{k+1}−b_k=−c^{−k} and \displaystyle \sum_{n=1}^∞\frac{k}{c^k}=a_1b_1+\frac{1}{c−1}\sum_{k=1}^∞c^{−k}=\frac{c}{(c−1)^2}.
3) \displaystyle \sum_{n=1}^∞\frac{n^2}{2^n}
4) \displaystyle \sum_{n=1}^∞\frac{(n+1)^2}{2^n}
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Solution: \displaystyle 6+4+1=11
Exercise \PageIndex{8}
The kth term of each of the following series has a factor \displaystyle x^k. Find the range of \displaystyle x for which the ratio test implies that the series converges.
1) \displaystyle \sum_{k=1}^∞\frac{x^k}{k^2}
2) \displaystyle \sum_{k=1}^∞\frac{x^{2k}}{k^2}
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Solution: \displaystyle |x|≤1
3) \displaystyle \sum_{k=1}^∞\frac{x^{2k}}{3^k}
4) \displaystyle \sum_{k=1}^∞\frac{x^k}{k!}
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Solution: \(\displaystyle |x|<∞\
5) Does there exist a number \displaystyle p such that \displaystyle \sum_{n=1}^∞\frac{2^n}{n^p} converges?
6) Let \displaystyle 0<r<1. For which real numbers \displaystyle p does \displaystyle \sum_{n=1}^∞n^pr^n converge?
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Solution: All real numbers \displaystyle p by the ratio test.
7) Suppose that \displaystyle \lim_{n→∞}∣\frac{a_{n+1}}{a_n}∣=p. For which values of \displaystyle p must \displaystyle \sum_{n=1}^∞2^na_n converge?
8) Suppose that \displaystyle \lim_{n→∞}∣\frac{a_{n+1}}{a_n}∣=p. For which values of \displaystyle r>0 is \displaystyle \sum_{n=1}^∞r^na_n guaranteed to converge?
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Solution: \displaystyle r<1/p
9) Suppose that \displaystyle ∣\frac{a_{n+1}}{a_n}∣ ≤(n+1)^p for all \displaystyle n=1,2,… where \displaystyle p is a fixed real number. For which values of \displaystyle p is \displaystyle \sum_{n=1}^∞n!a_n guaranteed to converge?
10) For which values of \displaystyle r>0, if any, does \displaystyle \sum_{n=1}^∞r^{\sqrt{n}} converge? (Hint: \displaystyle sum_{n=1}^∞a_n=\sum_{k=1}^∞\sum_{n=k^2}^{(k+1)^2−1}a_n.)
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Solution: \displaystyle 0<r<1. Note that the ratio and root tests are inconclusive. Using the hint, there are \displaystyle 2k terms \displaystyle r^\sqrt{n} for \displaystyle k^2≤n<(k+1)^2, and for \displaystyle r<1 each term is at least \displaystyle r^k. Thus, \displaystyle \sum_{n=1}^∞r^{\sqrt{n}}=\sum_{k=1}^∞\sum_{n=k^2}^{(k+1)^2−1}r^{\sqrt{n}} ≥\sum_{k=1}^∞2kr^k, which converges by the ratio test for \displaystyle r<1. For \displaystyle r≥1 the series diverges by the divergence test.
11) Suppose that \displaystyle ∣\frac{a_{n+2}}{a_n}∣ ≤r<1 for all \displaystyle n. Can you conclude that \displaystyle \sum_{n=1}^∞a_n converges?
12) Let \displaystyle a_n=2^{−[n/2]} where \displaystyle [x] is the greatest integer less than or equal to x. Determine whether \displaystyle \sum_{n=1}^∞a_n converges and justify your answer.
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Solution: One has \displaystyle a_1=1, a_2=a_3=1/2,…a_{2n}=a_{2n+1}=1/2^n. The ratio test does not apply because \displaystyle a_{n+1}/a_n=1 if \displaystyle n is even. However, \displaystyle a_{n+2}/a_n=1/2, so the series converges according to the previous exercise. Of course, the series is just a duplicated geometric series.
Exercise \PageIndex{9}
The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if \displaystyle \lim_{n→∞}\frac{a_{2n}}{a_n}<1/2, then \displaystyle \sum a_n converges, while if \displaystyle \lim_{n→∞}\frac{a_{2n+1}}{a_n}>1/2, then \displaystyle \sum a_n diverges.
1) Let \displaystyle a_n=\frac{1}{4}\frac{3}{6}\frac{5}{8}⋯\frac{2n−1}{2n+2}=\frac{1⋅3⋅5⋯(2n−1)}{2^n(n+1)!}. Explain why the ratio test cannot determine convergence of \displaystyle \sum_{n=1}^∞a_n. Use the fact that \displaystyle 1−1/(4k) is increasing \displaystyle k to estimate \displaystyle \lim_{n→∞}\frac{a_{2n}}{a_n}.
2) Let \displaystyle a_n=\frac{1}{1+x}\frac{2}{2+x}⋯\frac{n}{n+x}\frac{1}{n}=\frac{(n−1)!}{(1+x)(2+x)⋯(n+x).} Show that \displaystyle a_{2n}/a_n≤e^{−x/2}/2. For which \displaystyle x>0 does the generalized ratio test imply convergence of \displaystyle \sum_{n=1}^∞a_n? (Hint: Write \displaystyle 2a_{2n}/a_n as a product of \displaystyle n factors each smaller than \displaystyle 1/(1+x/(2n)).)
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Solution: \displaystyle a_{2n}/a_n=\frac{1}{2}⋅\frac{n+1}{n+1+x}\frac{n+2}{n+2+x}⋯\frac{2n}{2n+x}. The inverse of the \displaystyle kth factor is \displaystyle (n+k+x)/(n+k)>1+x/(2n) so the product is less than \displaystyle (1+x/(2n))^{−n}≈e^{−x/2}. Thus for \displaystyle x>0, \frac{a_{2n}}{a_n}≤\frac{1}{2}e^{−x/2}. The series converges for \displaystyle x>0.
3) Let \displaystyle a_n=\frac{n^{lnn}}{(lnn)^n}. Show that \displaystyle \frac{a_{2n}}{a_n}→0 as \displaystyle n→∞.