Binomial series

a. Find the first four nonzero terms of the Taylor series centered at 0 for the given function.

b. Use the first four terms of the series to approximate the given quantity.

\(f(x)=(1+x)^{-2 / 3}\); approximate \(1.18^{-2 / 3}\).

Solution 33EStep 1:Given thatf(x)= (1 + x)2/3 approximate 1.182/3Step2:To finda. Find the first four nonzero terms of the Taylor series centered at 0 for the given function.b. Use the first four terms of the series to approximate the given quantity.Step 3:a. The first four nonzero terms of the Taylor series centered at 0 for the given function.We know that Taylor series of function f(x) at a is defined asFirst termf(a)= at point x=0 = 1Second term =Third Term =Fourth term =Therefore,the first four nonzero terms of the Taylor series centered at 0 for the given function are 1,,,Step4:b. The first four terms of the series to approximate the given quantityf(x)=1-+x=0.18f(0.18)=f(0.18)=1-+ =1-0.12+0.018-0.00288 =1.018-0.12288 =0.89512=0.89512