produce a general formula for the degree n taylor polynomials for the following functions all using a = 0 as the point of approximation 1. 1/(1-x) 2. sin(x)
Produce a general formula for Taylor polynomials with degree-n as the following functions using a=0. Please step by step!(1) y(x)=1/(1-x) => y(0)=1 y'(x)=-1/(1-x)^2 => y'(0)=-1y"=2/(1-x)^3 => y"(0)=2y"'=-6/(1-x)^4 => y"'(0)=-3!y<4>(x)=24/(1-x)^5 => y<4>(0)=4!..................y<n>(x)=(-1)^n*n!/(1-x)^(n+1) => y<n>(0)=(-1)^n*n!y(x)=y(0)+y'(0)x+y"(0)x^2/2!+y"'(0)x^3/3!+.....=1-x+2!*x^2/2!-3!x^3/3!+.....=1-x+x^2-x^3+x^4-x^5+x^6-......
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Produce a general formula for Taylor polynomials with degree-n as the following functions using a=0. Please step by step!(1) y(x)=1/(1-x) => y(0)=1 y'(x)=-1/(1-x)^2 => y'(0)=-1y"=2/(1-x)^3 => y"(0)=2y"'=-6/(1-x)^4 => y"'(0)=-3!y<4>(x)=24/(1-x)^5 => y<4>(0)=4!..................y<n>(x)=(-1)^n*n!/(1-x)^(n+1) => y<n>(0)=(-1)^n*n!y(x)=y(0)+y'(0)x+y"(0)x^2/2!+y"'(0)x^3/3!+.....=1-x+2!*x^2/2!-3!x^3/3!+.....=1-x+x^2-x^3+x^4-x^5+x^6-......
(2) y(x)=sin(x), y(0)=0y'(x)=cos(x), y'(0)=1y"(x)=-sin(x), y"(0)=0y"'(x)=-cos(x), y"'(0)=-1y""(x)=sin(x), y""(0)=0y<5>(x)=cos(x), y<5>(0)=1............y(x)=y(0)+y'(0)x+y"(0)x^2/2!+y"'(0)x^3/3!+.....=0+x+0-x^3/3!+0+x^5/5!+0-x^7/7!+.....=x-x^3/3!+x^5/5!-x^7/7!+.....