Show Work Explain Find the Maclaurin series for the functio

Show Work & Explain.

Find the Maclaurin series for the function f (x) = In (1 + x) and deduce from the result the Maclaurin series of g (x) = In (1 - x).

Solution

f(x) = a0 + a1(x) + a2 x*x /2! ...... take derivative to find coeffecients : : put x=0 : a0 = 0 take first derivative : 1/(1+x) = a1 : and put x=0 : a1 = 1 second derivative : -1/(1+x)^2 = a2 and put x=0 : a2 = -1 similary : a3 = 2 a4 = -6 . f(x) = x - x*x/2! + 2*x*x*x/3! - 3! x*X*x*X/4! .... f(x) = x - x*x/2! + x*x*x/3! - x*x*x*x/4! .......... for getting ln(1-x) replace x by -x : g(x) = - x - x*x/2! - x*x*x/3! - x*x*x*x/4! .......... well range starts from n=1 : n= infinity : A) option fits the best but range should be from n=1 to n= infinity then only it gives proper answer !!!