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mathematics
precalculus
Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions
Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a.ƒ(x) = sin x, a = 0
Find the first four terms of the binomial series for the function.(1 + x)1/3
Which of the series defined by the formulas converge, and which diverge? Give reasons for your answers. X n=1 an
Use the Comparison Test to determine if each series converges or diverges. 1 Σ n=2 √n 1
Use the Ratio Test to determine if each series converges absolutely or diverges. 8 (n = 1)! Σ n=1 (n + 1)²
(a) Find the series’ radius and interval of convergence. For what values of x does the series converge (b) Absolutely, (c) Conditionally? Σ(-1)"(4x + 1)" n=0
Use substitution (as in Example 4) to find the Taylor series at x = 0 of the functions. EXAMPLE 4 Using known series, find the first few terms of the Taylor series for the given function using power series operations. (b) e cos x (a) (2x (2x + x cos x) (a) (2x (2x + x cos x) = (b) e cos x = = 2 = 1
Find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges. 1-2 +48 + + (−1)n-12n-1 ... + ..
Gives a formula for the nth term an of a sequence {an}. Find the values of a1, a2, a3, and a4. an = (-1)2+1 2n - 1
Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. 4 (In n)2 Σ(-1)", n=2
Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each convergent sequence. an 1 - 2n 2n
Use the Comparison Test to determine if each series converges or diverges. n + 2 Σ ₂2 n=2 n² – 1
Use the Ratio Test to determine if each series converges absolutely or diverges. 2n+1 Σ n=1n3n-1
Use the Integral Test to determine if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. n=1 1 " +4
What theorems are available for calculating limits of sequences? Give examples.
(a) Find the series’ radius and interval of convergence. For what values of x does the series converge (b) Absolutely, (c) Conditionally? 8 n=1 (3x 2)" n
Which of the series defined by the formulas converge, and which diverge? Give reasons for your answers. X n=1 an
Gives a formula for the nth term an of a sequence {an}. Find the values of a1, a2, a3, and a4. an 2 + (-1)" =
Find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges. 1 + 2.3 1 + 3.4 1 4.5 + : + (n + + 1 1)(n + 2)
Use substitution (as in Example 4) to find the Taylor series at x = 0 of the functions.cos 5x2 EXAMPLE 4 Using known series, find the first few terms of the Taylor series for the given function using power series operations. (b) e cos x (a) (2x (2x + x cos x) (a) (2x (2x + x cos x) = (b) e cos x
Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a.ƒ(x) = ln x, a = 1
Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each convergent sequence. an = 1 + (0.9)
Find the first four terms of the binomial series for the function.(1 - x)-3
Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. n Σ(1)". n² + 1 n=1
Use the Comparison Test to determine if each series converges or diverges. cosa n n=1_n3/2 Σ M8
(a) Find the series’ radius and interval of convergence. For what values of x does the series converge (b) Absolutely, (c) Conditionally? 8 n=0 (x - 2)" 10n
What are the six commonly occurring limits in Theorem 5 that arise frequently when you work with sequences and series? THEOREM 5-The Net Change Theorem The net change in a differentiable function F(x) over an interval a ≤ x ≤ b is the integral of its rate of change: F(b) F(a) - Sº = F'(x)
Use the Integral Test to determine if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. 8 -2n Lemn n=1
Use the Ratio Test to determine if each series converges absolutely or diverges. 8 Σ n=1 nt (-4)"
What theorem sometimes enables us to use l’Hôpital’s Rule to calculate the limit of a sequence? Give an example.
Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each convergent sequence. = an sin NTT 2
Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a.ƒ(x) = ln (1 + x), a = 0
Gives a formula for the nth term an of a sequence {an}. Find the values of a1, a2, a3, and a4. an 2n 2n+1
Which of the series defined by the formulas converge, and which diverge? Give reasons for your answers. X n=1 an
Use substitution (as in Example 4) to find the Taylor series at x = 0 of the functions.cos (x2/3/√2) EXAMPLE 4 Using known series, find the first few terms of the Taylor series for the given function using power series operations. (b) e cos x (a) (2x (2x + x cos x) (a) (2x (2x + x cos x) = (b) e
Find the first four terms of the binomial series for the function.(1 - 2x)1/2
Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. Σ(-1)+1 n=1 n? + 5 n² + 4
Find the first four terms of the binomial series for the function. 1 + X -2
Find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges. 5 1.2 + 5 + 2.3 5 3.4 + + 5 n(n + 1) +
(a) Find the series’ radius and interval of convergence. For what values of x does the series converge (b) Absolutely, (c) Conditionally? Σ (2x)" n=0
Use the Ratio Test to determine if each series converges absolutely or diverges. Σ n=2 3n+2 In n
Use the Comparison Test to determine if each series converges or diverges. ∞ n=1 1 n3n
Find the first four terms of the binomial series for the function. 1 بنانا 4
Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a.ƒ(x) = 1/x, a = 2
Gives a formula for the nth term an of a sequence {an}. Find the values of a1, a2, a3, and a4. an || 2" - 1 24
Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. Σ n=0 (-1)" 4n
Use substitution (as in Example 4) to find the Taylor series at x = 0 of the functions.ln (1 + x2) EXAMPLE 4 Using known series, find the first few terms of the Taylor series for the given function using power series operations. (b) e cos x (a) (2x (2x + x cos x) (a) (2x (2x + x cos x) = (b) e cos
Use the Integral Test to determine if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. :00 Σ n=1 n n? + 4 +4
Use the Comparison Test to determine if each series converges or diverges. Σ n=1 n + 4 +4 Vnt + 4
Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. Σ(-1)+1 n=1 ₂2 n²
(a) Find the series’ radius and interval of convergence. For what values of x does the series converge (b) Absolutely, (c) Conditionally? nxn Σ n=on + 2
What is an infinite series? What does it mean for such a series to converge? To diverge? Give examples.
Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each convergent sequence. an = In (n²) n
Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each convergent sequence.an = sin nπ
Use substitution (as in Example 4) to find the Taylor series at x = 0 of the functions.tan-1 (3x4) EXAMPLE 4 Using known series, find the first few terms of the Taylor series for the given function using power series operations. (b) e cos x (a) (2x (2x + x cos x) (a) (2x (2x + x cos x) = (b) e cos
Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. Σ 4n n=2
Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a.ƒ(x) = 1/(x + 2), a = 0
Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. 10n Σ(1)", (n + 1)! n=1
Use the Comparison Test to determine if each series converges or diverges. M8 Vn + 1 + 3 Σ n=1 \ Vn
(a) Find the series’ radius and interval of convergence. For what values of x does the series converge (b) Absolutely, (c) Conditionally? n=1 (-1)^(x + 2)" n
Use the Integral Test to determine if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. ∞ n=2 In (n²) n
Use substitution (as in Example 4) to find the Taylor series at x = 0 of the functions. EXAMPLE 4 Using known series, find the first few terms of the Taylor series for the given function using power series operations. (b) e cos x (a) (2x (2x + x cos x) (a) (2x (2x + x cos x) = (b) e cos x = = 2 = 1
Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each convergent sequence. an In (2n + 1) n
Use the Ratio Test to determine if each series converges absolutely or diverges. n5n Σ (2n + 3) In (n + 1) n=1
Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a.ƒ(x) = sin x, a = π/4
Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. 7 Σ 4" n=1
Find the first four terms of the binomial series for the function.(1 + x3)-1/2
Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. n Σ(-1)+1 (1) 00 n=1 n n
Taylor’s formulaexpresses the value of ƒ at x in terms of the values of ƒ and its derivatives at x = a. In numerical computations, we therefore need a to be a point where we know the values of ƒ and its derivatives. We also need a to be close enough to the values of ƒ we are interested in to
Use the Limit Comparison Test to determine if each series converges or diverges. ∞ Σ n=1 n – 2 n³n² + 3
Use the Integral Test to determine if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. ∞ n² n=1 en/3
Besides geometric series, what other convergent and divergent series do you know?
Gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.a1 = 1, an+1 = an/(n + 1)
(a) Find the series’ radius and interval of convergence. For what values of x does the series converge (b) Absolutely, (c) Conditionally? ∞ xn Σ n=inVn3n η n=1η 3η
Use the Root Test to determine if each series converges absolutely or diverges. ME 7 Σ (2n + 5)" n=1
Find the first four terms of the binomial series for the function.(1 + x2)-1/3
Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each convergent sequence. an n + Inn n
Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. 1 In n Σ(-1)+1, n=2
Find the first four terms of the binomial series for the function. 1 + X 1/2
Use the Integral Test to determine if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. 200 n - 4 Σ. n=n? – 2n + 1
Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. Σ(1)"; 4n n=0
Use the Limit Comparison Test to determine if each series converges or diverges. Σ n=1 n Vn? + 1 + 2
Gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.a1 = 2, an+1 = (-1)n+1an/2
Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a.ƒ(x) = √x, a = 4
(a) Find the series’ radius and interval of convergence. For what values of x does the series converge (b) Absolutely, (c) Conditionally? n=1 (x - 1)n √n
Taylor’s formulaexpresses the value of ƒ at x in terms of the values of ƒ and its derivatives at x = a. In numerical computations, we therefore need a to be a point where we know the values of ƒ and its derivatives. We also need a to be close enough to the values of ƒ we are interested in to
Use the Root Test to determine if each series converges absolutely or diverges. ∞ 4n ηξη (3η)" Σ
Which of the sequences whose nth terms appear converge, and which diverge? Find the limit of each convergent sequence. an || In (2n³ + 1) n
Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. M8 5 2" n Σ + n=0 3n
Find the first four terms of the binomial series for the function. X 1 + x
Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. Σ(-1)+1 n=1 Inn n
(a) Find the series’ radius and interval of convergence. For what values of x does the series converge (b) Absolutely, (c) Conditionally? Σ n=0 (−1)"x" n!
Taylor’s formulaexpresses the value of ƒ at x in terms of the values of ƒ and its derivatives at x = a. In numerical computations, we therefore need a to be a point where we know the values of ƒ and its derivatives. We also need a to be close enough to the values of ƒ we are interested in to
What can be said about term-by-term sums and differences of convergent series? About constant multiples of convergent and divergent series?
Which of the series converge, and which diverge? Give reasons for your answers. n=1 1 10n
Use the Limit Comparison Test to determine if each series converges or diverges. n(n + 1) Σ n=2 (n? + 1)(n – 1)
Use the Root Test to determine if each series converges absolutely or diverges. Σ n=1 ( 3η n 4η + 3 5 T
Gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.a1 = -2, an+1 = nan/(n + 1)
Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a.ƒ(x) = √1 - x, a = 0
Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. Σ(-1)"In (1 ( n=1 1 + 1 n
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