Homework Soursefx lnxx a1 Solution sory i solve for a5 Recall that the Tay
f(x) = ln(x)/x a=1
Solution
sory i solve for a=5 Recall that the Taylor series of a polynomial centered at x = a is: f(a) + f\'(a)(x - a)/1! + f\'\'(a)(x - a)^2/2! + f\'\'\'(a)(x - a)^3/3! + ... We obtain that: f(x) = ln(x) ==> f(5) = ln(5) f\'(x) = 1/x ==> f\'(5) = 1/5 f\'\'(x) = -1/x^2 ==> f\'\'(5) = -1/25 f\'\'(x) = 2/x^3 ==> f\'\'\'(5) = 2/125 Then, the first 4 non-zero terms of the series is: f(a) + f\'(a)(x - a)/1! + f\'\'(a)(x - a)^2/2! + f\'\'\'(a)(x - a)^3/3! = ln(5) + (x - 5)/5 + (x - 5)^2/50 - 2(x - 5)^3/750
