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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Determine the limit of the sequence or show that the sequence diverges.an = 21/n2
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ n=1 40 n n!
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = x2ex2
Determine the limit of the sequence or show that the sequence diverges.an = 10n/n!
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = ex−2
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ n=0 n! 6n
Determine the limit of the sequence or show that the sequence diverges.bm = 1 + (−1)m
Find the Maclaurin series and find the interval on which the expansion is valid. f(x) = 1 - cos x
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ 6n n=0 n!
Determine the limit of the sequence or show that the sequence diverges. bm 1 + (-1) m
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = ln(1 − 5x)
Determine the limit of the sequence or show that the sequence diverges. bn = tan-1 n+2 n+ 5
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 1 Σ (2η)! n=1
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = (x2 + 2x)ex
Determine the limit of the sequence or show that the sequence diverges. an 100" n! 3 + π" 5n
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ n=1 n? (2n + 1)!
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = sinh x
Determine the limit of the sequence or show that the sequence diverges.bn = √n2 + n − √n2 + 1
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 n=1 (n!)³ (3n)!
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = cosh x
Determine the limit of the sequence or show that the sequence diverges.cn = √n2 + n − √n2 − n
Use the Direct Comparison Test to determine whether the infinite series is convergent. 00 n=1 1 n¹/3 + 2n
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = ex sin x
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. ∞0 Σ n=2 1 2n + 1
Determine the limit of the sequence or show that the sequence diverges. Cп = [1 + 3 n n
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = ex ln(1 − x)
Assume that |an+1/an| converges to ρ = 1/3. What can you say about the convergence of the given series? 00 Σnan n=1
Use series to determine a reduced fraction that has decimal expansion 0.121212 · · · .
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 n=2 1 In n
Determine the limit of the sequence or show that the sequence diverges. Cп 1 + n 3 n
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary. f(x) = sin x 1- x
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary. f(x) = 1 1 + sin x
Determine the limit of the sequence or show that the sequence diverges.bn = ln(n + 1) − ln n)
Show that converges if|x| 00 Σημ n n=1
Determine the limit of the sequence or show that the sequence diverges. Cn In(n2 + 1) In(n3 + 1) -
Use the Squeeze Theorem to show that lim n→∞ arctan(n) √n 0.
Show thatconverges if |x| 00 n=1 2″ Τ n
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = (1 + x)1/4
Show that converges for all r. 00 n=1 ph n!
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = (1 + x)−3/2
Give an example of a divergent sequence {an} such that {sin an} is convergent.
Show that converges if |r| 00 n=1 pn n
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = ex tan−1 x Using the Maclaurin series for ex and tan−1 x, we find.
Calculate where an = 1/2 3n − 1/32n. an+1 lim n→∞⁰ an
Define an+1 = √an + 6 with a1 = 2.(a) Compute an for n = 2, 3, 4, 5.(b) Show that {an} is increasing and is bounded by 3.(c) Prove that exists and find its value. lim an n→∞0
Is there any value of k such that converges? 00 n=1 2n nk
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = sin (x3 − x)
Calculate the partial sums S4 and S7 of the series 00 n=1 n-2 n² + 2n
The following limit could be helpful: lim 1+ n→∞0 n n = e.
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = esin x
The following limit could be helpful: lim 1+ n→∞0 n n = e.
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = e(ex)
Find the sum 4/9 + 8/27 + 16/81 + 32/243 + · · · .
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = cosh(x2)
Assume that |an+1/an| converges to ρ = 1/3. What can you say about the convergence of the given series? 00 Σ n=1 H 2 an
Find the terms through degree 4 of the Maclaurin series of ƒ(x). Use multiplication and substitution as necessary.ƒ(x) = sinh(x) cosh(x)
Assume that |an+1/an| converges to ρ = 1/3. What can you say about the convergence of the given series? 00 Σ n=1 3" dn
Use series to determine a reduced fraction that has decimal expansion 0.108108108 · · · .
Find the Taylor series centered at c and the interval on which the expansion is valid. f(x) = 1 c=1
Assume that |an+1/an| converges to ρ = 1/3. What can you say about the convergence of the given series? 00 n=1 4' an
Find the sum2n+3/3n. 00 Σ n=-1
Find the Taylor series centered at c and the interval on which the expansion is valid. ƒ(x) = e3x, c = −1
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
Assume that |an+1/an| converges to ρ = 1/3. What can you say about the convergence of the given series? 8 ∞0 2 Σα n n=1
Find the Taylor series centered at c and the interval on which the expansion is valid. f(x) = 1 1- x' c=5
Show thatdiverges if b ≠ π/2. 00 -1 Σ ( – tan-' n') n=1
Assume that |an+1/an| converges to ρ = 4. Does converge (assume that an ≠ 0 for all n)? 00 n=1 an
Find the Taylor series centered at c and the interval on which the expansion is valid. f(x) = sin x, c = π 2
Give an example of divergent series 00 00 Σan and Σb, such that Σ(an + bn) = 1. n=1 n=1 n=1
Find the Taylor series centered at c and the interval on which the expansion is valid. ƒ(x) = x4 + 3x − 1, c = 2
Let Compute SN for N = 1, 2, 3, 4. Find S by showing that S = IS 1 n+ 2
Find the Taylor series centered at c and the interval on which the expansion is valid. ƒ(x) = x4 + 3x − 1, c = 0
Use the Root Test to determine convergence or divergence (or state that the test is inconclusive). ∞0 1 Σ 10n n=0
Find the Taylor series centered at c and the interval on which the expansion is valid. f(x) = 1 x2 c = 4
Evaluate S = 8 Σ n=3 1 n(n + 3)
Use the Root Test to determine convergence or divergence (or state that the test is inconclusive). 00 n=1 1 nn
Find the total area of the infinitely many circles on the interval [0, 1] in Figure 1. 0 -100 114 -X
Find the Taylor series centered at c and the interval on which the expansion is valid. ƒ(x) = √x, c = 4
Use the Root Test to determine convergence or divergence (or state that the test is inconclusive). ∞ Σ k=0 k k 3k + 1/
Use the Root Test to determine convergence or divergence (or state that the test is inconclusive). 00 n=4 1 + n -n²
Use the Direct Comparison or Limit Comparison Test to determine whether the infinite series converges. 00 Σ n=1 1 √n+n
Find the interval of convergence. 00 Σe"(x – 2)" n=12
Determine whether the series converges absolutely, conditionally, or not at all. ∞0 (−1) Σ (0.999)" n=0
Determine convergence or divergence by any method. ك n=1 3" + (-2)" 5n
Find the interval of convergence. 00 Σ(-1)'n³ (x-7)" n=1
Find the interval of convergence. 00 Σ27" (x − 1)3n+2 n=0
Find the interval of convergence. 8 00 Σ n=1 2n =(x + 3)" 3n
Determine convergence or divergence by any method. M8 Σ n=1 1 3n4 + 12n
Determine convergence or divergence by any method. 00 Σ n=l (−1)" Th2 + 1
Determine convergence or divergence by any method. 8 00 1 Σ n=1_Vn2 + 1
Use the Integral Test to determine whether the infinite series is convergent. n=1 1 n+ 3
Use the Integral Test to determine whether the infinite series is convergent. 00 Σ n=1 -1/3 n
Use the Integral Test to determine whether the infinite series is convergent. 00 1 Σ n=5 Vn-4
Use the Integral Test to determine whether the infinite series is convergent. 8 n Σ (n? + 9)5/2 n=25
Either use the formula for the sum of a geometric series to find the sum, or state that the series diverges. Σ(3) n 4
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a. f(x) = sinx, a = KIN π 2
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a.ƒ(x) = sin x, a = 0
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a. f(x) = 1 1 + x a = 2
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a. f(x) = 1 1 + x²⁹ a = -1
Calculate the Taylor polynomials T2 and T3 centered at x = a for the given function and value of a. f(x) = x² +1 x+1' a = -2

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