Find the Taylor series for f centered at 7 if (-1)^n! 3n (n + 1) f(n) (7) = n = 0 What is the radius of convergence R of the Taylor series? R =

Which option below represents the Taylor series for the function x 0? about 1+ 2x4 a) 0 E-1)" - 2" xn+2 n=0 2(-1)--1 2x4+2 n! b) n=0 2" x 4n+2 (-1)". n! n=0

(-1)^n! 1) Suppose we know that f(n) (4) = and the Taylor series of f 3n (n+1) centered at 4 converges to f(x) for all x in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates f(5) with error less than 0.0002.

Evaluate the infinite series by identifying it as the value of an integral of a geometric series. (– 1)" 87 +1(n + 1) 2 In (3) x n=0 1 n+1 |f(t)dt se)dt where f(=) = (- )"(;) 1 x" = Hint: Write it as 8 8 + x n=0