For the following function, find the full power series centered at a of convergence. 0 and then give the first 5 nonzero terms of the power series and the open interval = f(2) Σ 8 1(x)--(-1)*(3)* n=0 ₤(x) = + + + ++... The open interval of convergence is: 1 1 3 f(x)= = 28 3x6 +1 (Give your answer in help (intervals) .)

Find the power series (centered at zero) of the function f ( x ) = 3 /4 + x2 and state the interval of convergence. Write the first terms of the power series. You do not have to express the series in sigma notation. Don't forget to state the interval of convergence.

Find the radius and interval of convergence for the following series: (x - 2)" 10" a. n=0 Radius= 10 Interval = b. Σ n=0 Radius= (-1)" xn n! (-8,12) C. Interval= (-∞0,00) 8 Σ(n!) (x-4)" n=0 Radius= Interval=

Please select only the CONDITONALLY CONVERGENT ones. Thank you!

Determine the interval of convergence of the power series: > n=1 (-)" (7 – 1)" (x- 1)" n 3 A. A single point x = 1 3D1 B. -2

Find the radii of convergence for each of the following power series. (a) (b) (c) n=1 n(x - 1)" 2n (n-1)x", n=1 Hint: The radius of convergence is 1. Explain why you can show this by checking that the series converges for x = -1, but diverges for x = 1. To show the latter, it is helpful to recall that (1+1)* S.M. ≤n. n=2 (x - π)" n(n − 1)*

Find all x for which the power series converges. Sigmak=1 to infinity (x-3)^k/k