Binomial series a. Find the first four nonzero terms of the binomial 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)23, approximate (1.02)23

Question

Binomial series  a. Find the first four nonzero terms of the binomial 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)23, approximate (1.02)23

Laith Petty · Accepted Answer

Given function is f(x)=(1+x)23  For function f(x)=(1+x)n binomial series centered at 0 is given as:  (1+x)n=1+nx+n(n−1)2!x2+n(n−1)(n−2)3!x3+...  We have, f(x)=(1+x)23  Here, n=23  Putting n=23 in the series, we get  (1+x)23=1+23x+23(23−1)2x2+23(23−1)(23−2)6x3+...  =1+23x+23⋅−132x2+23⋅−13⋅−436x3+...  =1+23x−19x2+481x3+...  First four non-zero terms are:  (1+x)23=1+23x−19x2+481x3  Now, to approximate (1.02)23  (1.02)23=(1+0.02)23  Put x=0.02 in the series we obtained above.  (1+0.02)23=1+23⋅(0.02)−19⋅(0.02)2+481⋅(0.02)3  =1+0.043−0.00049+481⋅(0.000008)  =1+0.0133−0.000044+0.000000392  =1.0128

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Binomial series a. Find the first four nonzero terms of the binomial 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)^{frac{2}{3}}, approximate (1.02)^{frac{2}{3}}

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