6. [-/0.11 Points] DETAILS SCALCET9 11.XP. 10.001. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R (x) - 0.] f( x ) = sin (- ) A(x ) = I 1=0 Find the associated radius of convergence R. R = Need Help? Read it Watch It 7. [-/0.11 Points] DETAILS SCALCET9 11.XP. 10.004. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) - 0.] (( x ) = x4 - 4x2 + 4, a =2 f ( x ) = Find the associated radius of convergence R. RE Need Help? Read It Watch It 8. [-/0.11 Points] DETAILS SCALCET9 11.10.024.MI. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) - 0.] f ( x ) = 6 , a = -4 F( X ) = I n = 0 Find the associated radius of convergence R. R : Need Help? Read it Master It 9. [-/0.11 Points] DETAILS SCALCET9 11.10.035. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Use the binomial series to expand the function as a power series. V 1 - x O1- 1x - 3 . 7 ... (4n - 5) n = 2 47 . n! o (-)nti(4n - 5)" on n = 0 40 O1+* + 3 . 7 ... (4n - 5 ) n n = 2 n! O1 - 1x - 3 . 7 ... (4n - 5) n n = 2 n! O1+ (-in+13 . 7 ... (4n - 5) n = 1 4" . n! State the radius of convergence, R. R=3. [-/0.11 Points] DETAILS SCALCET9 11.10.AE.003. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Example Video Example () Prove that 5ex is equal to the sum of its Maclaurin series. Solution If f(x) = 5ex, then f(n + 1)(x) = for all n. If d is any positive number and |x| S d, then If ( n + 1) ( x )1 = S 5ed. So Taylor's Inequality, with a = 0 and M = 5ed, says that IR, (x) IS |xin+1 for | x/ s d. (n + 1)! Notice that the same constant M = 5e" works for every value of n. But, from this equation, we have 5ed lim _|xin+ = 5ed lim |x|7+ 1 7 -+ 0 (n + 1)! n - (n + 1)! It follows from the Squeeze Theorem that lim |R,(x) | = 0 and therefore lim R (x) = for all values of x. n - By this theorem, 5e* is equal to the sum of its Maclaurin series, that is, 5ex = \\ 5x7 for all x. n! Need Help? Read It1. [-/0.11 Points] DETAILS SCALCET9 11.10.AE.002. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Example Video Example ()) Find the Maclaurin series of the function f(x) = e2x and its radius of convergence. Solution If f(x) = e2x, then f(7) (x) = so f(") (0) = 2"e = for all n. Therefore the Taylor series for f at 0 (that is, the Maclaurin series) is given. 5 ()(0) *n = ) 20x n = 0 n! n = 0 2x = 1+_ 8x3 3 ! To find the radius of convergence we let a = - 20xn -. Then we have the following. n! an + 1 n! an 20xn (n + 1)! 21x| (K ) x" = 1 4 kx + K(k - 1) x2 + K(k - 1)(k - 2) 3 1 ... n = 0 2 ! 3! We know from the binomial series that this series converges when - ~|