DO EXAMPLE 1 For what values ofx is the series 2 n!x3" convergent? n = O SOLUTION We use the Ratio Test. If we let an, as usual, denote the nth term of the series, then an = n!x3". If X at O, we have Mimi a, | "m x naoo x x By the Ratio Test, the series diverges when x at 0. Thus the given series converges only when \"my Find a power series representation for the function. f(X)- arctan(X) z E n=0 Determine the radius of convergence, R. R = 9 Need Help? Use a Maclaurin series in the table below to obtain the Maclaurin series for the given function. f(x) = 3cos nix 4 00 f ( x ) = S n = 0 X IM: x" = l+ x+ x+x+... 1 R =1 X * = S X = 1+ + + R = 00 n-o n! 1! 3! 1 8 x2n+1 sin x = >(-1)" X + R = 00 m-0 (2n + 1)! 3! 5! 71 18 x 2n COS X = >(-1)" 1 - + + R = 00 (2n)! 21 4! 6! x 2n+1 tan 'x = >(-1)" = x + + R = 1 2n + 1 3 5 7 (1 + x) * = k x" = 1 + kx + k(k - 1) IM : + k(k - 1)(k - 2) 3. 3+ ... R =1 2! 3! :Enhanced Feedback Please try again, keeping in mind that you need to use the MacLaurin series for cos(x)