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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Use the Integral Test to determine whether the infinite series is convergent. 00 n=1 Inn n²
Forverify that for n ≥ 1, 00 n=1 1 n+ √n'
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
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
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
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
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 +
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
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,
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)
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
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
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
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
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
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)
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)
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
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
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
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
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
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
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)
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
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
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
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
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
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
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)|

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