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Mathematics LibreTexts

11: Sequences and Series (Exercises)

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These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

11.1: Sequences

Ex 11.1.1 Compute limxx1/x. (answer)

Ex 11.1.2 Use the squeeze theorem to show that limnn!nn=0.

Ex 11.1.3 Determine whether {n+47n}n=0 converges or diverges. If it converges, compute the limit. (answer)

Ex 11.1.4 Determine whether {n2+1(n+1)2}n=0 converges or diverges. If it converges, compute the limit. (answer)

Ex 11.1.5 Determine whether {n+47n2+3n}n=1 converges or diverges. If it converges, compute the limit. (answer)

Ex 11.1.6 Determine whether {2nn!}n=0 converges or diverges. (answer)

11.2: Series

Ex 11.2.1 Explain why n=1n22n2+1 diverges. (answer)

Ex 11.2.2 Explain why n=1521/n+14 diverges. (answer)

Ex 11.2.3 Explain why n=13n diverges. (answer)

Ex 11.2.4 Compute n=04(3)n33n. (answer)

Ex 11.2.5 Compute n=032n+45n. (answer)

Ex 11.2.6 Compute n=04n+15n. (answer)

Ex 11.2.7 Compute n=03n+17n+1. (answer)

Ex 11.2.8 Compute n=1(35)n. (answer)

Ex 11.2.9 Compute n=13n5n+1. (answer)

11.3: The Integral Test

Determine whether each series converges or diverges.

Ex 11.3.1 n=11nπ/4 (answer)

Ex 11.3.2 n=1nn2+1 (answer)

Ex 11.3.3 n=1lnnn2 (answer)

Ex 11.3.4 n=11n2+1 (answer)

Ex 11.3.5 n=11en (answer)

Ex 11.3.6 n=1nen (answer)

Ex 11.3.7 n=21nlnn (answer)

Ex 11.3.8 n=21n(lnn)2 (answer)

Ex 11.3.9 Find an N so that n=11n4 is between Nn=11n4 and Nn=11n4+0.005. (answer)

Ex 11.3.10 Find an N so that n=01en is between Nn=01en and Nn=01en+104. (answer)

Ex 11.3.11 Find an N so that n=1lnnn2 is between Nn=1lnnn2 and Nn=1lnnn2+0.005. (answer)

Ex 11.3.12 Find an N so that n=21n(lnn)2 is between Nn=21n(lnn)2 and Nn=21n(lnn)2+0.005. (answer)

11.4: Alternating Series

Determine whether the following series converge or diverge.

Ex 11.4.1 n=1(1)n12n+5 (answer)

Ex 11.4.2 n=4(1)n1n3 (answer)

Ex 11.4.3 n=1(1)n1n3n2 (answer)

Ex 11.4.4 n=1(1)n1lnnn (answer)

Ex 11.4.5 Approximate n=1(1)n11n3 to two decimal places. (answer)

Ex 11.4.6 Approximate n=1(1)n11n4 to two decimal places. (answer)

11.5: Comparison Test

Determine whether the series converge or diverge.

Ex 11.5.1 n=112n2+3n+5 (answer)

Ex 11.5.2 n=212n2+3n5 (answer)

Ex 11.5.3 n=112n23n5 (answer)

Ex 11.5.4 n=13n+42n2+3n+5 (answer)

Ex 11.5.5 n=13n2+42n2+3n+5 (answer)

Ex 11.5.6 n=1lnnn (answer)

Ex 11.5.7 n=1lnnn3 (answer)

Ex 11.5.8 n=21lnn (answer)

Ex 11.5.9 n=13n2n+5n (answer)

Ex 11.5.10 n=13n2n+3n (answer)

11.6: Absolute Convergence

Determine whether each series converges absolutely, converges conditionally, or diverges.

Ex 11.6.1 n=1(1)n112n2+3n+5 (answer)

Ex 11.6.2 n=1(1)n13n2+42n2+3n+5 (answer)

Ex 11.6.3 n=1(1)n1lnnn (answer)

Ex 11.6.4 n=1(1)n1lnnn3 (answer)

Ex 11.6.5 n=2(1)n1lnn (answer)

Ex 11.6.6 n=0(1)n3n2n+5n (answer)

Ex 11.6.7 n=0(1)n3n2n+3n (answer)

Ex 11.6.8 n=1(1)n1arctannn (answer)

11.7: The Ratio and Root Tests

Ex 11.7.1 Compute limn|an+1/an| for the series 1/n2.

Ex 11.7.2 Compute limn|an+1/an| for the series 1/n.

Ex 11.7.3 Compute limn|an|1/n for the series 1/n2.

Ex 11.7.4 Compute limn|an|1/n for the series 1/n.

Determine whether the series converge.

Ex 11.7.5 n=0(1)n3n5n (answer)

Ex 11.7.6 n=1n!nn (answer)

Ex 11.7.7 n=1n5nn (answer)

Ex 11.7.8 n=1(n!)2nn (answer)

Ex 11.7.9 Prove theorem 11.7.3, the root test.

11.8: Power Series

Find the radius and interval of convergence for each series. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence.

Ex 11.8.1 n=0nxn (answer)

Ex 11.8.2 n=0xnn! (answer)

Ex 11.8.3 n=1n!nnxn (answer)

Ex 11.8.4 n=1n!nn(x2)n (answer)

Ex 11.8.5 n=1(n!)2nn(x2)n (answer)

Ex 11.8.6 n=1(x+5)nn(n+1) (answer)

11.9: Calculus with Power Series

Ex 11.9.1 Find a series representation for ln2. (answer)

Ex 11.9.2 Find a power series representation for 1/(1x)2. (answer)

Ex 11.9.3 Find a power series representation for 2/(1x)3. (answer)

Ex 11.9.4 Find a power series representation for 1/(1x)3. What is the radius of convergence? (answer)

Ex 11.9.5 Find a power series representation for ln(1x)dx. (answer).

11.10: Taylor Series

For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence.

Ex 11.10.1 cosx (answer)

Ex 11.10.2 ex (answer)

Ex 11.10.3 1/x, a=5 (answer)

Ex 11.10.4 lnx, a=1 (answer)

Ex 11.10.5 lnx, a=2 (answer)

Ex 11.10.6 1/x2, a=1 (answer)

Ex 11.10.7 1/1x (answer)

Ex 11.10.8 Find the first four terms of the Maclaurin series for tanx (up to and including the x3 term). (answer)

Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for xcos(x2). (answer)

Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for xex. (answer)

11.11: Taylor's Theorem

Ex 11.11.1 Find a polynomial approximation for cosx on [0,π], accurate to ±103 (answer)

Ex 11.11.2 How many terms of the series for lnx centered at 1 are required so that the guaranteed error on [1/2,3/2] is at most 103? What if the interval is instead [1,3/2]? (answer)

Ex 11.11.3 Find the first three nonzero terms in the Taylor series for tanx on [π/4,π/4], and compute the guaranteed error term as given by Taylor's theorem. (You may want to use Sage or a similar aid.) (answer)

Ex 11.11.4 Show that cosx is equal to its Taylor series for all x by showing that the limit of the error term is zero as N approaches infinity.

Ex 11.11.5 Show that ex is equal to its Taylor series for all x by showing that the limit of the error term is zero as N approaches infinity.

11.12: Additional Exercises


This page titled 11: Sequences and Series (Exercises) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David Guichard via source content that was edited to the style and standards of the LibreTexts platform.

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