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 limx→∞x1/x. (answer)
Ex 11.1.2 Use the squeeze theorem to show that limn→∞n!nn=0.
Ex 11.1.3 Determine whether {√n+47−√n}∞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+47√n2+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)n−33n. (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+10−4. (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)n−12n+5 (answer)
Ex 11.4.2 ∑∞n=4(−1)n−1√n−3 (answer)
Ex 11.4.3 ∑∞n=1(−1)n−1n3n−2 (answer)
Ex 11.4.4 ∑∞n=1(−1)n−1lnnn (answer)
Ex 11.4.5 Approximate ∑∞n=1(−1)n−11n3 to two decimal places. (answer)
Ex 11.4.6 Approximate ∑∞n=1(−1)n−11n4 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+3n−5 (answer)
Ex 11.5.3 ∑∞n=112n2−3n−5 (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)n−112n2+3n+5 (answer)
Ex 11.6.2 ∑∞n=1(−1)n−13n2+42n2+3n+5 (answer)
Ex 11.6.3 ∑∞n=1(−1)n−1lnnn (answer)
Ex 11.6.4 ∑∞n=1(−1)n−1lnnn3 (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)n−1arctannn (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(x−2)n (answer)
Ex 11.8.5 ∑∞n=1(n!)2nn(x−2)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/(1−x)2. (answer)
Ex 11.9.3 Find a power series representation for 2/(1−x)3. (answer)
Ex 11.9.4 Find a power series representation for 1/(1−x)3. What is the radius of convergence? (answer)
Ex 11.9.5 Find a power series representation for ∫ln(1−x)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/√1−x (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 xe−x. (answer)
11.11: Taylor's Theorem
Ex 11.11.1 Find a polynomial approximation for cosx on [0,π], accurate to ±10−3 (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 10−3? 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.