Chapter 10, Problem 62RE

Find a power series representation for x2cos(2x) and then by differentiation, show that the equation below is correct. Please answer and full detail and write it clearly,

Find the power series for the functions below. Write your answers with the sum starting at n=0  a) f(x) = ln (x+1) b) r(x) = x^4/(1+2x)^3 ( 1+2x in the parenthesis raised to the power of 3).

Find the MacLaurin Series for f (x) = cos(2/x) (-1)"(2x)" (2n)! -x+ 등-. 3 = 1 – x + 6 90 .. (-1)"(4x)" (2n)! = 1-2x + 을x2-.. 2 2 4 Zn=0 45 .. (-1)"x2" (2n)! .2 = 1 – * + - X. Zn=0 2 24 720 .. (-1)"x" (2n)! = 2 – x + - 100 3 Zn=0 720 ..

(b) Now, lets evaluate the same integral using power series. First, find the power series for the function 24 f(z) Then, integrate it from 0 to 2, and call it S. S should be an infinite series. ² +4 What are the first few terms of S? ao = 12 aj = 12 12 a3 12 a4 = (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a)), you have found an estimate for the value of r in terms of an infinite series. Approximate the value of r by the first 5 terms. 1052 315 (d) What is the upper bound for your error of your estimate if you use the first 10 terms? (Use the alternating series estimation.) 4 23

with process/explanation

Use the power series X f(x) = (1-x)² Select one: a. c. x e. Ση·(-1)*·(x)" n = 1 x 1 1-X Ση·(x)+1 n = 1 En.(x)" n = 1 = d. Σn (-1)" (x)” +¹ n = 1 Σn. (x)n-1 n = 1 x Σx", x<1 to determine a power series for the function n=0

I need help on this. Thank you

= cosx to find first four terms Use the Maclaurin Series for w(x) = e*and q(x) for nonzero f (x) = w(x) · q(x) Write down the first four terms in the binomial series for (1 + 5x)-4

Verify that Jp+1(x) + 2J½ (x) – Jp-1(x) = 0 Tip: Compare the series of Jp+1(x) – Jp-1(x) and – 2J½(x) = 0