(H) The geometric series (i) Converge (i) Diverge

Real analysis

b and c

Find the power series for each function and identify its interval of convergence. f(x) = 1 – x2 1.

= (10 points) For the function g(x) = x = (1 + c) x + cx³, show that a 1 is a fixed point and determine for which values of the constant c the iteration sequence for initial conditions sufficiently close to a converge. For what values of c, if any, is the convergence quadratic? Verify your results graphically and numerically.

V2+cos (n²+1) Use the basic comparison test to determine whether E-1 converges or n+en diverges.

3- Find the binomial series of the function f(x) = (1– 2x)3

f(x) = f(x)= Σ n=0 6 x22x-8 1 4"+1 X + (-1)"+1) 2"+1 Find the interval of convergence. (Enter your answer using interval notation.) (-2,4]

(x - 1)n (n + 1)2n Determine the values of x for which the power series Σ(−1)”- n=0 converges. Identify the radius of convergence.

(d) Determine whether each of the following series converge or diverge, using any of the standard tests for divergence. (i) ΣΕ=1 2(-1)kk² k3-1 k³-1 (ii) k-12k²