1 6. Using the power series = Σ c" |x | < 1, find a power series about O for 1 х n=0 1 and state the radius of convergence. (2 - x)2
Use the equation 1 1x = Ž for 1x1 <1 1 - X n = 0 to expand the function in a power series with center c = 0. 1 f(x) 5 + x3 00 Σ n = 0 Determine the interval of convergence. (Enter your answer using interval otation.)
Use the power series 1 1 + X = Ë (-1)^x), 1x! < 1 n=0 to find a power series for the function, centered at 0. 1 g(x) x + 1 00 g(x) = Σ n=0 Determine the interval of convergence. (Enter your answer using interval notation.)
4. Use the power series representaion f(t) = In(1 - 1) =- for -1 <<1, k=1 to find the power series representation for the following function(centered at 0). Give the interval of convergence of the new series. p(r) = 2.r" ln(1-2) 5. Find the power series representation for g centered at 0 by differentiating or integrating the power series of f(perhaps more than once). Give the interval of convergence for the resulting series. 1 using (3) 1-
- Express the Fourier series of that particular function - when - 1 < x < 0 4 f(x) = { when 0<x<T
analyze the convergence/divergence of the next seriesυ) Σ* (ο <h <) 2=1 24 1) X vii) i) Σ η = 1 31 (α > 1, k 50 ) η=1 u?
Find the value of k such that: Pr(-k<Z<k) = 0.60
. c) + < 2 b) 2 + 3x 27, 0. Solve for r: r' + 2.r < 2.1? +12
Use the power series 1 1 + x = (-1) (-1)", IX1 < 1 no to find a power series for the function, centered at 0. g(x) = 1 9 x + 1 00 g(x) = Σ no Determine the interval of convergence. (Enter your answer using interval notation.) Submit Answer
Hint: use geometric series and the theorem on differentiation of a
power series
6.7 Obtain power series expansions for (1z <1. (Hint: use 6.11.) and for (1+z, each valid for l