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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Show that converges by using the Direct Comparison Test with 00 Σ n=2 1 \n2 – 3
Which of the following inequalities can be used to study the convergence of Explain. 00 1 Σ -? n2 + \n M=2
Use the Direct Comparison Test to determine whether the infinite series is convergent. 00 n=1 1 n2n
Use the Direct Comparison Test to determine whether the infinite series is convergent. Ins n=1 3 n n5 + 4n + 1
Use the Direct Comparison Test to determine whether the infinite series is convergent. Μ8 Σ n=1 1 √n³ + 2n-1
Use the Direct Comparison Test to determine whether the infinite series is convergent. 00 m=1 4 m! + 4m
Use the Direct Comparison Test to determine whether the infinite series is convergent. 00 n=4 √n n-3
Use the Direct Comparison Test to determine whether the infinite series is convergent. 00 k=1 sin² k k²
Use the Direct Comparison Test to determine whether the infinite series is convergent. k=2 k2/9 k10/91
Use the Direct Comparison Test to determine whether the infinite series is convergent. 00 n=1 2 3n+3-n
Use the Direct Comparison Test to determine whether the infinite series is convergent. 00 Στ k=1 2-4²
Use the Direct Comparison Test to determine whether the infinite series is convergent. 00 n=1 1 (n + 1)!
Use the Direct Comparison Test to determine whether the infinite series is convergent. 00 n! n=1 است n³
For all a > 0 and b > 1, the inequalities ln n ≤ na, na n are true for n sufficiently large (this can be proved using L’Hˆopital’s Rule). Use this, together with the Direct Comparison Test, to determine whether the series converges or diverges. 00 n=1 In n 3 n
For all a > 0 and b > 1, the inequalities ln n ≤ na, na n are true for n sufficiently large (this can be proved using L’Hˆopital’s Rule). Use this, together with the Direct Comparison Test, to determine whether the series converges or diverges. m=2 1 In m
For all a > 0 and b > 1, the inequalities ln n ≤ na, na n are true for n sufficiently large (this can be proved using L’Hˆopital’s Rule). Use this, together with the Direct Comparison Test, to determine whether the series converges or diverges. n=1 (Inn) 100 1.1 n
For all a > 0 and b > 1, the inequalities ln n ≤ na, na n are true for n sufficiently large (this can be proved using L’Hˆopital’s Rule). Use this, together with the Direct Comparison Test, to determine whether the series converges or diverges. 00 1 Σ n=1 (Inn)10
Show that converges. Use sin x ≤ x for x ≥ 0. 00 n=1 1 sin - -1/2
Use the Limit Comparison Test to prove convergence or divergence of the infinite series. 00 n=2 n² n4 - 1
Find a formula for the general term an (not the partial sum) of the infinite series. (a) (c) (d) - + - 3 9 27 1 2² 1 2.1 + 81 33 3.2.1 2 1 + 1²+1 2² +1 + 44 4.3.2.1 2 3² + 1 + + 1 4² + 1 (b) + 5|2 + 25 125 + 4 8
Write in summation notation: (a) 1+ (c) 1 (d) - 125 9 1 4 3 + + 1 - 5 - 625 16 16 1 7 + +... 3125 25 + 15,625 36 + ... (b) 1 1 + + 16 25 1 36
Compute the partial sums S2, S4, and S6. 1 + 1 2² + 1 3² + 1 4² +
Compute the partial sums S2, S4, and S6. 00 Σ(-1)*και k=1
Compute the partial sums S2, S4, and S6. 1 + 1.2 1 2.3 + 1 3.4
Compute the partial sums S2, S4, and S6. 00 1 j=1 - 15 j!
The series 1 +(1/5) + (1/5)2 + (1/5)3 + · · · converges to 5/4. Calculate SN for N = 1, 2, . . . until you find an SN that approximates 5/4 with an error less than 0.0001.
Use a computer algebra system to compute S10, S100, S500, and S1000 for the series. Do these values suggest convergence to the given value? 5 4 3 = 1 2.3.4 1 4.5.6 1 6.7.8 1 8.9.10
The series 1/0! − 1/1! + 1/2! − 1/3! + · · · is known to converge to e−1 (recall that 0! = 1). Calculate SN for N = 1, 2, . . . until you find an S/N that approximates e−1 with an error less than 0.001.
Use a computer algebra system to compute S10, S100, S500, and S1000 for the series. Do these values suggest convergence to the given value? 90 || 1 + 1 24 + 1 34 1 44
Calculate S3, S4, and S5 and then find the sum of the telescoping series 00 n=1 1 n+1 1 n+ 2
Writeas a telescoping series and find its sum. 00 Σ n=3 1 n(n − 1)
Use partial fractions to rewriteas a telescoping series and find its sum. 00 1 In(n + 3) n=1
Find a formula for the partial sum SN of and show that the series diverges. 00 SN of (-1)"-1 n=1
Use the nth Term Divergence Test (Theorem 4) to prove that the following series diverge. THEOREM 4 If f is continuous and lim an = 81x = L, then lim fan)=f(lim an) 11-0 In) = . ƒ(L) In other words, we may pass a limit of a sequence inside a continuous function. 815
Find the sum of 1/1 · 3 + 1/3 · 5 + 1/ 5 · 7 + · · · .
Use the nth Term Divergence Test (Theorem 4) to prove that the following series diverge. 00 THEOREM 4 nth Term Divergence Test If lim an # 0, then the series Σan #=1 diverges.
Use the nth Term Divergence Test (Theorem 4) to prove that the following series diverge. 00 THEOREM 4 nth Term Divergence Test If lim an # 0, then the series Σan #=1 diverges.
Use the nth Term Divergence Test (Theorem 4) to prove that the following series diverge. 00 THEOREM 4 nth Term Divergence Test If lim an # 0, then the series Σan #=1 diverges.
Use the nth Term Divergence Test (Theorem 4) to prove that the following series diverge. 00 THEOREM 4 nth Term Divergence Test If lim an # 0, then the series Σan #=1 diverges.
Use the nth Term Divergence Test (Theorem 4) to prove that the following series diverge. 00 THEOREM 4 nth Term Divergence Test If lim an # 0, then the series Σan #=1 diverges.
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 1 + - 1 + 6 36 216 |- + 1 +
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 43 53 44 45 + 54 55
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 7-3 7 7 + 32 + 7 33 + 7 34 +...
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 7 + + +
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. Σ(1) -n
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. n=2 7.(-3)" 5n
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 00 Μ Σ(3) n=0
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 00 n=1 -n
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. n=0 3(-2)" - 5" 8n
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 00 Σε n=2 3-2n
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 5- 1 5 5 5 + 4 4² 43 +
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 00 n=0 8+2" n 5n
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. + + 26 74
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 7 49 343 + 64 512 100 8 2401 4096 +
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. 25 5 + +1+ 9 3 3 5 + 9 25 + 27 + 125
Show, by giving counterexamples, that the assertions of Theorem 1 are not valid if the series an and b, are not convergent. n=0 n=0
Let an = 1/2n − 1 for n = 1, 2, 3, . . . . Write out the first three terms of the following sequences. (a) bn = an+1 (c) dn = an (b) Cn = an+3 (d) en = 2an - An+1
Calculate the first four terms of the sequence, starting with n = 1. Cп || 3″ n!
Calculate the first four terms of the sequence, starting with n = 1. bn (2n-1)! n!
Calculate the first four terms of the sequence, starting with n = 1. b₁ = 1, bn =bn-1 + 1 bn-1
Calculate the first four terms of the sequence, starting with n = 1.a1 = 2, an+1 = 2a2n − 3
Calculate the first four terms of the sequence, starting with n = 1.bn = 5 + cos πn
Calculate the first four terms of the sequence, starting with n = 1. Cn = 1 + 1 + 2 3 - + ... + 1 n
Calculate the first four terms of the sequence, starting with n = 1.cn = (−1)2n+1
Calculate the first four terms of the sequence, starting with n = 1. W, = 1 + 1 22 - 32 + 1 n² -
Calculate the first four terms of the sequence, starting with n = 1. b₁ = 2, b₂ = 3, bn = 2bn-1 + bn-2
Calculate the first four terms of the sequence, starting with n = 1. an Fn+1 Fn where Fn is the nth Fibonacci number.
Find a formula for the nth term of each sequence. (a) 1 -1 1' 8' 27' (b) 2 3 4 - 6' 7' 8
Suppose that Determine: lim an n→∞o = = 4 and lim bn = 7. n→∞
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges.an = 5 − 2n THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an = f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an= f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an = f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an = f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an= f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an = f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges.cn = 9n THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an = f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges.zn = 10−1 THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an = f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an= f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an = f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 4 to determine the limit of the sequence. THEOREM 4 If f is continuous and lim a, L, then = 11-0 lim f(an) = f(lim an) = f(L) In other words, we may pass a limit of a sequence inside a continuous function.
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an= f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges.rn = ln n − ln(n2 + 1) THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an = f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an= f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 1 to determine the limit of the sequence or state that the sequence diverges.yn = ne1 THEOREM 1 Sequence Defined by a Function If lim f(x) exists, then the se- XIX quence an = f(n) converges to the same limit: lim an = lim f(x) 818 xx
Use Theorem 4 to determine the limit of the sequence. THEOREM 4 If f is continuous and lim a,= L, then 11-0 lim f(an) = flim an) = f(L) In other words, we may pass a limit of a sequence inside a continuous function.
Use Theorem 4 to determine the limit of the sequence.an = e4n/(3n+9) THEOREM 4 If f is continuous and lima, = L, then 11-0 lim f(an) = f(lim an) = f(L) In other words, we may pass a limit of a sequence inside a continuous function.
Use Theorem 4 to determine the limit of the sequence.an = tan−1(e−n) THEOREM 4 If f is continuous and lim a, = L, then 11-0 lim f(an) = f(lim an) = f(L) In other words, we may pass a limit of a sequence inside a continuous function.
Let an = Fn+1 / Fn, where {Fn} is the Fibonacci sequence. The sequence {an} has a limit. We do not prove this fact, but investigate the value of the limit in these exercises.Estimate to five decimal places by computing an for sufficiently large n. lim an n- 900
Let an = n + 1. Find a number M such that: (a) Jan 1| ≤ 0.001 for n ≥ M. (b) lan 1 ≤ 0.00001 for n ≥ M. - Then use the limit definition to prove that lim an = 1. n-∞
Let bn =(1/3)n. (a) Find a value of M such that bn ≤ 10-5 for n ≥ M. (b) Use the limit definition to prove that lim bn = 0. n→∞
Let an = Fn+1 / Fn, where {Fn} is the Fibonacci sequence. The sequence {an} has a limit. We do not prove this fact, but investigate the value of the limit in these exercises.Denote the limit of {an} by L. Given that the limit exists, we can determine L as follows:(a) Show that an+1 = 1 + 1
Use the Limit Comparison Test to prove convergence or divergence of the infinite series. Σ n=2 1 n(n – 1)
Use the Root Test to determine convergence or divergence (or state that the test is inconclusive). 00 k=0 (416) (k + 10 k k
Prove the following variant of the Alternating Series Test: If {bn} is a positive, decreasing sequence with then the seriesconverges.Show that S3N is increasing and bounded by a1 + a2, and continue as in the proof of the Alternating Series Test. lim bn = 0, n→∞
Use Eq. (2) to expand the function in a power series with center c = 0 and determine the interval of convergence. 1-x Σ #=0 for [x] < 1
Find the Taylor series centered at c and the interval on which the expansion is valid. f(x) = 1 1-x²³ c = 3
Use the Integral Test to determine whether the infinite series converges. 00 Σ n=1 τη n η3 + 1
Compute the Taylor polynomial indicated and use the Error Bound to find the maximum possible size of the error. Verify your result with a calculator.ƒ(x) = cos x, a = 0; |cos 0.25 − T5(0.25)|
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. an 10+ (;) п
Use the Limit Comparison Test to prove convergence or divergence of the infinite series. n=2 n √n³ + 1

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