00 ar For Ir| < 1, Σn(ar" = ar + 2(ar)? + 3(ar) + ... = Use this to find the power series for f(x) = (1 – ar)? (1+x)² * n=1 Select the correct answer below: 00 O 2 n(x)" n=1 00 o 2n(-x)²n n=1 00 (-x)" n=1 00 E n(-x)" n=1 (х)" n=1 O 2-nx)" n=1

1 g(x) = (7 +x)3 g(x) = 2 5 „n (-1)"(n+1)(n + 2) 2.(7)"+3 n=0 Part 3 Use your answers above to now express the function as a power series (centered at x = 0). x2 h(x) = (7 + x)3 h(x) = n=0

(2n+ 1)x Please explan Determine whether the series sin- converges or diverg es. If it converges, find its sum. thish L and how it nelotes to the n-D0 (4r + 1)x Since sin =1 and sin =-1 and the angle may be incremente by any multiple of 2x, then sin 2 =1 and (4r+3)a sin =-1, (any r). How does this nelate to the answer. Examine the partial sums of the series. The expanded series is sin 3n 5x + sin + sin ,+ sin + sin+ .. ansver Notice that the angles in consecutive even terms (n is even) differ by 2x, and the angles in consecutive odd terms (n is odd) differ by 2x. Since sin 0 = sin (0+2mx), all the even terms are equal to 1, and all the odd terms are equal to -1. Evaluating each term, the series is 1+(-1)+1+(-1)+1+ The partial sums altermate between 1 and 0, tending to (2n + 1)a neither value. Therefore, lim sin- k00 n=0 does not exist and the series diverges. detail. Pelease explain n thanke gou.

Use a power series to represent the following function: f(x)=x^2/2-x