Use the power series 1 1 + x f(x) = n = 0 to find a power series for the function, centered at 0. f(x) = In(x + 1) 80 n = 0 ∞ (−1)"x", |x| < 1 X Determine the interval of convergence. (Enter your answer using interval notation.) (-1,1)

V12(-1)" 3" (2n + 1)" 1. Consider the series n=0 (a) Use any test for convergence/divergence to show that the series converges. V12(-1)" 3" (2n + 1) (b) It is possible to show that the sum of the series is T, in other words, the series n=0 converges to the number T. (You do NOT need to prove this, but it can be done somewhat easily using a Taylor series expansion of arctan x.) Suppose you want to use a partial sum of this series to estimate the value of T to an accuracy of within 0.0001. Would using the first 8 terms of the series be enough to ensure you get an accuracy of within 0.0001? (8 terms means the terms where n = 0, 1, 2, 3, ..., 7.) Hint: Use Theorem 5.14.

It keeps saying my interval notation or notation is wrong

00 1 Use the equation 2x" for |x| < 1 to expand the function in a power series with center c = 0. 1 - x 1- x4 n=0 (Express numbers in exact form. Use symbolic notation and fractions where needed.) Σ 1- x n=0 Determine the interval of convergence. (Give your answer as an interval in the form (*,*). Use the symbol co for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Express numbers in exact form. Usc symbolic notation and fractions where needed.)