Taylor series centered at \(a \neq 0\)

a. Find the first four nonzero terms of the Taylor series for the given function centered at a.

b. Write the power series using summation notation.

f(x) = In x, a = 3

Solution 21EStep 1:a) In this problem we need to find first four nonzero terms of the taylor series for the function f(x) = ln(x) , centered at a = 3 .We know that , the taylor series of the function centered at ‘a’ is : f(x) = f(a) + (x-a)++Given ; f(x) = ln(x), centered at a = 3. f(x) = , then f(3) = ln(3) , then , since (ln(x)) = , then , since = , then = , then ………………Therefore , the taylor series of the function f(x) = ln(x), centered at a = 3is ; f(x) = ln(x) = f(3) +(x-3)++ = ln(3)+ (x-3)++ = ln(3) +(x-3)-+Therefore , the first four nonzero terms of the taylor series centered at ‘3’ for the function ’ are ln(3) +(x-3) -+That is , ln(3) ,(x-3) , -,First term = ln(3)Second term = (x-3)Third term = -Fourth term =