Josephood7

 

Find the power series representation for f(x)=∫x0tan−1ttdt. f(x)=∑n=1∞(−1)enanxpn, where en= A. n - 1 B. n C. 0 and an= 1/((2*n-1)**2) , and pn= Find the sum of ∑∞n=1n(n+1)xn= The function f(x)=10(1+10x)2 is represented as a power series: f(x)=∑n=0∞cnxn Find the first few coefficients in the power series. c0= 10 c1= -200 c2= 3000 c3= -40000 c4= 500000 Find the radius of convergence R of the series. Use Eq. (1) from the text to expand the function into a power series with center c=0 and determine the set of x for which the expansion is valid. f(x)=18+x6 18+x6=∑n=0∞ (a) Evaluate the integral ∫2040x2+4dx. Your answer should be in the form kπ, where k is an integer. What is the value of k? (Hint: darctan(x)dx=1x2+1 ) k= 5 (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x)=40x2+4. Then, integrate it from 0 to 2, and call it S. S should be an infinite series. What are the first few terms of S ? a0= 20 a1= -20/3 a2= 20/5 a3= -20/7 a4= 20/9 (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 π in terms of an infinite series. Approximate the value of π by the first 5 terms. (20-20/3+20/5-20/7+20/9)/5 . (d) What is the upper bound for your error of your estimate if you use the first 7 terms? (Use the alternating series estimation.) 20/15/5 . Find the sum of the following series. If it is divergent, type "Diverges" or "D". −∑n=1∞1n(−14)n=14−132+1192−11024+⋯ Answer: