When x < 0, the series for e^x is an alternating series. Use the Alternating Series Estimation Theorem to estimate the error that results from replacing e^x by 1 + x + x^2/2 when -0.1 < x < 0. | Error |= (Round to four decimal places as needed.)
a. Use Taylor's formula with n = 2 to find the quadratic approximation of f(x) = (1+x)^k at x = 0 (k a constant). b. If k = 4, for approximately what values of x in the interval [0, 1] will the error in the quadratic approximation be less than 1/100? a. What is the quadratic approximation of f(x) = (1+x)^k at x = 0? Q(x) = b. If k = 4, for approximately what values of x in the interval [0, 1] is the error less than 1/100? A. 0 ≤ x < 25^-1/3 B. 0 ≤ x ≤ 1 C. 0 ≤ x < 800^-1/3 D. 0 ≤ x < 100^-1/3
Use the identity sin^2x = 1 - cos(2x)/2 to obtain the Maclaurin series for sin2x. Then differentiate this series to obtain the Maclaurin series for 2sinxcosx. Check that this is the series for sin(2x). What is the Maclaurin series for sin2x? What is the Maclaurin series for 2sinxcosx = sin(2x)?
Use the Alternating Series Estimation Theorem or Taylor’s Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. (Enter your answer using interval notation. Round your answers to three decimal places.) cos(x) ≈ 1 – x^2/2 + x^4/24 (|error |< 0.0005)
Use the Alternating Series Estimation Theorem or Taylor’s Formula to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. (Round your answers to three decimal places.) sin x ≈ x – x^3/6 (|error |< 0.01)
Define Q as the region that is bounded by the graph of the function g(y) = 2√y + 2, the y-axis, y = 1, and y = 2. Use the disk method to find the volume of the solid of revolution when Q is rotated around the y axis. Submit an exact answer in terms of π. Provide your answer below: V = units^3
Define Q as the region that is bounded by the graph of the function g(y) = -2√y + 2, the y-axis, y = 1, and y = 4. Use the disk method to find the volume of the solid of revolution when Q is rotated around the y axis. Submit an exact answer in terms of π. Provide your answer below: V = units^3
Define Q as the region that is bounded by the graph of the function g(y) = 2√y + 3, the y-axis, y = -1, and y = 2. Use the disk method to find the volume of the solid of revolution when Q is rotated around the y axis. Submit an exact answer in terms of π. Provide your answer below: V = units^3
Define Q as the region that is bounded by the graph of the function g(y) = 2√y + 4, the y-axis, y = -3, and y = -1. Use the disk method to find the volume of the solid of revolution when Q is rotated around the y axis. Submit an exact answer in terms of π. Provide your answer below: V = units^3
Define Q as the region that is bounded by the graph of the function g(y) = 2√y + 1, the y-axis, y = 3, and y = 8. Use the disk method to find the volume of the solid of revolution when Q is rotated around the y axis. Submit an exact answer in terms of π. Provide your answer below: V = units^3
Use the Root Test to determine if the following series converges absolutely or diverges. ∑ ∞ n=1 (-1)^n 1 – 8/n)^n2 ( Hint: limn→∞ (1 + x/n)^n = e^x ) Since the limit resulting from the Root Test is (Type an exact answer.)
Find the Maclaurin series for the function. coshx = e^x + e^-x/2 The Maclaurin series for the function is ∑n=0
Determine whether the alternating series ∑n=1 ∞ (-1)^n+1 4^n/n^4 converges or diverges. Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a geometric series with r = B. The series does not satisfy the conditions of the Alternating Series Test but converges because the limit used in the Root Test is C. The series converges by the Alternating Series Test. D. The series does not satisfy the conditions of the Alternating Series Test but diverges because the limit used in the Ratio Test is E. The series does not satisfy the conditions of the Alternating Series Test but diverges because it is a p-series with r =
Use the Ratio Test to determine if the following series converges absolutely or diverges. ∑ ∞ n=1 16^n/n! Since the limit resulting from the Ratio Test is
Find the Taylor series generated by f at x = a. f(x) = 2^x, a = 1 A. 2 x = ∑n=0 ∞ 2(x-1) n(ln2) n n! B. 2 x = ∑n=0 ∞ 2(x-1) n (ln 2) nn! c. 2 x = ∑n=0 ∞ 2(x-1) n+1 (ln2) n n! D. 2 x = ∑n=0 ∞ 2(x-1) n(ln2) n+1 n!
Use substitution to find the Taylor series at x = 0 of the function e^-14x. What is the general expression for the nth term in the Taylor series at x = 0 for e^-14x? ∑ ∞ n=0 (Type an exact answer.)
Does the sequence {an} converge or diverge? Find the limit if the sequence is convergent. an = ln(n + 1)/6√n Select the correct choice below and, if necessary, fill in the answer box to complete the choice. A. The sequence converges to limn→∞an = . (Simplify your answer.) B. The sequence diverges.
Use power series operations to find the Taylor series at x = 0 for the following function. 13xe^x The Taylor series for e^x is a commonly known series. What is the Taylor series at x = 0 for e^x? ∑ ∞ n=0 (Type an exact answer.) Use power series operations and the Taylor series at x = 0 for e^x to find the Taylor series at x = 0 for the given function. ∑ ∞ n=0 (Type an exact answer.)
Does the series ∑n=1 ∞ (-1)^n/√n + √n+17 converge absolutely, converge conditionally, or diverge? Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series converges absolutely because the limit used in the Root Test is B. The series diverges because the limit used in the nth-Term Test does not exist. C. The series diverges per the Comparison Test with ∑1 ∞ 1/√n. D. The series converges conditionally per the Alternating Series Test and because the limit used in the Root Test is not less than or equal to 1. E. The series converges absolutely because the limit used in the Ratio Test is F. The series converges conditionally per the Alternating Series Test and the Integral Test because ∫1 ∞ f(x)dx does not exist.
Write out the first few terms of the series ∑n=0 ∞ (4/3^n + (-1)^n/5^n ). What is the series' sum? The first term is . (Type an integer or a simplified fraction.) The second term is . (Type an integer or a simplified fraction.) The third term is . (Type an integer or a simplified fraction.) The fourth term is . (Type an integer or a simplified fraction.) If the series is convergent, what is the series' sum? Select the correct choice below and fill in any answer boxes within your choice. A. The series converges. The series' sum is (Type an integer or a simplified fraction.) B. The series diverges.
Use series to approximate the value of the integral with an error of magnitude less than 10^-8 ∫ 0.17 0 sinx/x dx ∫ 0.17 0 sinx/x dx ≈ (Round to eight decimal places as needed.)
(a) Find the series' radius and interval of convergence. Find the values of x for which the series converges (b) absolutely and (c) conditionally ∑ ∞ n=0 x^n/√n^2 + 11 (a) The radius of convergence is (Simplify your answer.) Determine the interval of convergence. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The interval of convergence is (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The series converges only at x = (Type an integer or a simplified fraction.) c. The series converges for all values of x. (b) For what values of x does the series converge absolutely? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The series converges absolutely for (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The series converges absolutely at x =. (Type an integer or a simplified fraction.) c. The series converges absolutely for all values of x. (c) For what values of x does the series converge conditionally? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The series converges conditionally for (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The series converges conditionally at x =. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) c. There are no values of x for which the series converges conditionally.
Use an appropriate test to determine whether the series given below converges or diverges. ∑ ∞ n=1 1/3^n-1 + 9 Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The Comparison Test with ∑ (1/3)^n-1 shows that the series converges. B. The series diverges because the limit used in the nth-Term Test is (Type an exact answer.) c. The Comparison Test with ∑ (1/3)^n-1 shows that the series diverges.
Does the series ∑n=1 ∞ (-1)^n+1 (0.3)^n converge absolutely, converge conditionally, or diverge? Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series converges conditionally since the corresponding series of absolute values is a geometric series with r = B. The series converges absolutely since the corresponding series of absolute values is a geometric series with r = c. The series diverges per the nth-Term Test. D. The series converges absolutely since the corresponding series of absolute values is a p-series with p =. E. The series converges conditionally since the corresponding series of absolute values diverges, but the series passes the Alternating Series Test.
Find the values of x for which the given geometric series converges. Also, find the sum of the series (as a function of x) for those values of x. ∑ ∞ n=0 sinn x Find the values of x for which the given geometric series converges. Choose the correct answer below. A. x ≠ kπ, k an integer B. |x| < 1/2 c. x ≠ (2k + 1) π/2, k an integer ∑n=0 ∞ sinn x =