18. [-/0.11 Points] DETAILS SCALCET9 11.10.068. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Use series to evaluate the limit. 1 - cos(2x) lim x - 0 1 + 2x - e2x Need Help? Read It 19. [-/0.11 Points] DETAILS SCALCET9 11.XP. 10.024.MI. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Find the sum of the series. E (-1)167x8n n = 0 n! Need Help? Read It Watch It Master It 20. [-/0.18 Points] DETAILS SCALCET9 11.10.087. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Find the sum of the series. 5(-1)"x 2n + 1 n = 0 22n + 1(2n + 1)! 4.3301 Need Help? Read It Watch It Viewing Saved Work Revert to Last Response10. [-/0.11 Points] DETAILS SCALCET9 11.10.037. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Use the binomial series to expand the function as a power series. 3 ( 6+ x) 3 State the radius of convergence, R. R = Need Help? Read It Watch It 11. [-/0.11 Points] DETAILS SCALCET9 11.XP. 10.015.MI. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Use a Maclaurin series in this table to obtain the Maclaurin series for the given function. f(x) = 8ex + e8x n =0 Need Help? Read It Watch It Master It 12. [-/0.11 Points] DETAILS SCALCET9 11.10.043.MI. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Use a Maclaurin series in this table to obtain the Maclaurin series for the given function. f(x) = 9x cos (_x2) E n = 0 Need Help? Read It Watch It Master It 13. [-/0.11 Points] DETAILS SCALCET9 11.10.047. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Use a Maclaurin series in this table to obtain the Maclaurin series for the given function. f(x) = 5 sin2(x) Hint: Use sin2(x) = = (1 - cos(2x)) E n = 1 Need Help? Read It Watch It15. [-/0.11 Points] DETAILS SCALCET9 11.10.AE.013. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Example Video Example () (a) Evaluate the integral below as an infinite series. e-4x2 (b) Evaluate the integral below correct to within an error of 0.001. .0.5 e -4x2 Solution (a) First we find the Maclaurin series for f(x) = e-4%". Although it's possible to use the direct method, let's find it simply by replacing x with -4x2 in the series for ex given in Table 1 in the book. Thus for all values of x, e -4x2 = 5 (-4x-)" = (-1)0 40x 20 n = 0 n! n = 0 n! = 1 - 4x2 64x6 1! 3! 2! Now we integrate term by term. Je-ax ax = / 1- 4x2 64x6 + ... + (-1)" dx 1! 31 2! n (-1)040 = C+ x- 16x5 64x7 7 . 3 ! . . . 5 . 2! . n! This series converges for all x because the original series for e-4% converges for all x. (b) The fundamental Theorem of Calculus gives e-4x2 dx = [x - 16x5 64x7 10.5 5 . 2 ! 7 . 3 ! - 1/6 + - 1/84 + 1/432 -... - 1/6 + - 1/84 + 1/432 = The Alternating Series Estimation Theorem shows that the error involved in this approximation is less than 0.001. Need Help? Read It 16. [-/0.11 Points] DETAILS SCALCET9 11.XP. 10.020.MI. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Evaluate the indefinite integral as an infinite series. sex - 1 dx E + c n = 1 Need Help? Read It Master It 17. [-/0.11 Points] DETAILS SCALCET9 11.10.061. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Evaluate the indefinite integral as an infinite series. cos(x) - 1 dx + c