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(a) Show that the function defined by the power series 20+1 y=(-1)" 2n +1 n=0 satisfies...
n=0 4. Using the power series cos(x) = { (-1)",2 (-0<x<0), to find a power (2n)!...
n=0 4. Using the power series cos(x) = { (-1)",2 (-0<x<0), to find a power (2n)! series for the function f(x) = sin(x) sin(3x) and its interval of convergence. 23 Find the power series representation for the function f(2) and its interval (3x - 2) of convergence. 5. +
1. Given the series -1)" n! , 2n+1 (2n1) (i) Find the radius of convergence of...
1. Given the series -1)" n! , 2n+1 (2n1) (i) Find the radius of convergence of the series. (ii) Find also the largest open interval on which the series converges. 2. (a) Find the Taylor series, in summation form, of f(x) = 1+1 (b) (i) (ii) Find the radius of convergence of the series. Find also the largest open interval on which the series converges. 3. (a) Find two series solutions of the differential equation +9=0, -oo < x <...
What 2n 7. Determine the radius and interval of convergence of the power series function has...
What 2n 7. Determine the radius and interval of convergence of the power series function has this power series as its Taylor series at 07 (10) 27-1 8. Consider the rational function (x) Find the Taylor series at 0 of (2) and determine its radius and interval of convergence. (10) 2-1
Use the binomial series to expand the function as a power series. f (x) = 5/1+...
Use the binomial series to expand the function as a power series. f (x) = 5/1+ -5/1+ 6 15(-1)*+1 (0) 2n! IM n=0 00 5 5+ 12+ + [51-1)^-1 (a)" 2n! n=2 72 5+ =1041.32... (2n – 1) () 72 5+ 5(-1)"1.3.5. .... (2n - 3) 2n! n=2 (2n – 3) 72 5 5+ - + 12" 5(-1)n-11.3.5.... 2n! n=2 State the radius of convergence, R. (If the radius of convergence is infinity, enter INFINITY.) R = X Need Heln2...
11 . 12 13 Find a power series representation for the function. (Give your power series...
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Find a power series representation for the function. (Give your power series representation centered at x = 0.) f(x) = 5 x f(x) = ;- n = 0 Determine the interval of convergence. (Enter your answer using interval notation.) Find a power series representation for the function. (Give your power series representation centered at x = 0.) x2 f(x) X4 + 16 f(x) = Σ |(-1)" ) n = 0 X (a) Use differentiation to find...
number 4 1. Find the limit of the following sequences (find lim an) n n a.)...
number 4
1. Find the limit of the following sequences (find lim an) n n a.) an = n +3 b.) an = V35n n- 2. Determine whether the following series converge or diverge. -3 (n + 2)n + 5 b.) tan-'(n) n2 + 1 a.) 5 nel 3. Determine the radius of convergence and the interval of convergence of the series 2" (x – 3)" n n=1 n=0 (-1)", 2n 4. Using the power series cos(x) (2n)! (-« <...
Find the interval of convergence of the power series: 5) 00 2n -(4x – 8)" n...
Find the interval of convergence of the power series: 5) 00 2n -(4x – 8)" n n=1 E (n + 1)(x - 2)" (2n + 1)! n=0 7) 00 w n(x + 10)" (2n)! n=0
Question 3 (1 point) The function f is defined by the power series 1)2 3! 5! 72n+1)! 1)% n-0 (2n+...
Question 3 (1 point) The function f is defined by the power series 1)2 3! 5! 72n+1)! 1)% n-0 (2n+1) ! for all real numbers x. Use the first and second derivative test by finding f(x) and f"(x). Determine whether f has a local maximum, a local minimum, or neither at x=0. Give a reason for your answer. Use the
Question 3 (1 point) The function f is defined by the power series 1)2 3! 5! 72n+1)! 1)% n-0 (2n+1)...
Find the interval of convergence for the given power series. (2 - 4)" 00 n=1 nl...
Find the interval of convergence for the given power series. (2 - 4)" 00 n=1 nl - 3)" The series is convergent from 2 = , left end included (enter Yor N): right end included (enter Y or N): to C = CI" 10.2 Suppose that (14 + 2) n=0 Find the first few coefficients. Со = C1 C2 C3 C4 Find the radius of convergence R of the power series. R= 2 The function f(x) is represented as a...
4. (a) Solve the differential equation (1-12)y"-2cy' + λ(A + 1)y 0 using power series centered at 0 , in which...
4. (a) Solve the differential equation (1-12)y"-2cy' + λ(A + 1)y 0 using power series centered at 0 , in which λ is a constant. Write your solution as a linear combination of two independent solutions whose coefficients are expressed in terms of λ . Compute the coefficients of each solution up to and including the 5 term. Without computing them, what is the smallest possible value of the radius of convergence of each solution and why? (b) When λ...
n=0 4. Using the power series cos(x) = { (-1)",2 (-0<x<0), to find a power (2n)! series for the function f(x) = sin(x) sin(3x) and its interval of convergence. 23 Find the power series representation for the function f(2) and its interval (3x - 2) of convergence. 5. +
1. Given the series -1)" n! , 2n+1 (2n1) (i) Find the radius of convergence of the series. (ii) Find also the largest open interval on which the series converges. 2. (a) Find the Taylor series, in summation form, of f(x) = 1+1 (b) (i) (ii) Find the radius of convergence of the series. Find also the largest open interval on which the series converges. 3. (a) Find two series solutions of the differential equation +9=0, -oo < x <...
What 2n 7. Determine the radius and interval of convergence of the power series function has this power series as its Taylor series at 07 (10) 27-1 8. Consider the rational function (x) Find the Taylor series at 0 of (2) and determine its radius and interval of convergence. (10) 2-1
Use the binomial series to expand the function as a power series. f (x) = 5/1+ -5/1+ 6 15(-1)*+1 (0) 2n! IM n=0 00 5 5+ 12+ + [51-1)^-1 (a)" 2n! n=2 72 5+ =1041.32... (2n – 1) () 72 5+ 5(-1)"1.3.5. .... (2n - 3) 2n! n=2 (2n – 3) 72 5 5+ - + 12" 5(-1)n-11.3.5.... 2n! n=2 State the radius of convergence, R. (If the radius of convergence is infinity, enter INFINITY.) R = X Need Heln2...
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Find a power series representation for the function. (Give your power series representation centered at x = 0.) f(x) = 5 x f(x) = ;- n = 0 Determine the interval of convergence. (Enter your answer using interval notation.) Find a power series representation for the function. (Give your power series representation centered at x = 0.) x2 f(x) X4 + 16 f(x) = Σ |(-1)" ) n = 0 X (a) Use differentiation to find...
number 4
1. Find the limit of the following sequences (find lim an) n n a.) an = n +3 b.) an = V35n n- 2. Determine whether the following series converge or diverge. -3 (n + 2)n + 5 b.) tan-'(n) n2 + 1 a.) 5 nel 3. Determine the radius of convergence and the interval of convergence of the series 2" (x – 3)" n n=1 n=0 (-1)", 2n 4. Using the power series cos(x) (2n)! (-« <...
Find the interval of convergence of the power series: 5) 00 2n -(4x – 8)" n n=1 E (n + 1)(x - 2)" (2n + 1)! n=0 7) 00 w n(x + 10)" (2n)! n=0
Question 3 (1 point) The function f is defined by the power series 1)2 3! 5! 72n+1)! 1)% n-0 (2n+1) ! for all real numbers x. Use the first and second derivative test by finding f(x) and f"(x). Determine whether f has a local maximum, a local minimum, or neither at x=0. Give a reason for your answer. Use the
Question 3 (1 point) The function f is defined by the power series 1)2 3! 5! 72n+1)! 1)% n-0 (2n+1)...
Find the interval of convergence for the given power series. (2 - 4)" 00 n=1 nl - 3)" The series is convergent from 2 = , left end included (enter Yor N): right end included (enter Y or N): to C = CI" 10.2 Suppose that (14 + 2) n=0 Find the first few coefficients. Со = C1 C2 C3 C4 Find the radius of convergence R of the power series. R= 2 The function f(x) is represented as a...
4. (a) Solve the differential equation (1-12)y"-2cy' + λ(A + 1)y 0 using power series centered at 0 , in which λ is a constant. Write your solution as a linear combination of two independent solutions whose coefficients are expressed in terms of λ . Compute the coefficients of each solution up to and including the 5 term. Without computing them, what is the smallest possible value of the radius of convergence of each solution and why? (b) When λ...
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