Geometric series can be found by transforming f(x)= to series form using 1 4 Maclaurin's series to get 1-X = E=0x² = 1 + x + x² + x³ + x¹ + 1) Why this series diverges at x = 4 while the function is defined. 2) Why this series diverges at x = -4 while the function is defined.

(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.

1 Start from the power series of the function g(x) = and then from 1-x there, step by step get to the power series expansion of the function x²+x Show the steps and write the final answer in summation (1-x)³ f(x) = =