x 6 6 2 1 3 x 2 2 x 4 4 x 6 6 3 x 2 2 1 4 1 2 1 2 2 x 4 1 6 1 2 2 1 2 1 4 1 3 1

X 6 6 2 1 3 x 2 2 x 4 4 x 6 6 3 x 2 2 1 4 1 2 1 2 2 x

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+x66!-...2-13x22!-x44!+x66!-...3+......=-x22!+14!-12·122x4+-16!-12·2·12!·-14!-13·123x6+.......=-12x2-124x4-145x6+....H. Find Taylor series forf(x) =1x+2atx= 0.Solution:Since11-x=n=0xnfor|x|<1, so that11+x=n=0(-1)nxnfor|x|<1. Thenf(x) =1x+ 2=12·11 +x2=12Xn=0(-1)nx2n=Xn=0(-1)n2n+1xn,∀|x|<2.I. Find Taylor series forf(x) =1x+2atx= 3.Solution:f(x) =1x+ 2=1x-3 + 5=15·11 +x-35=15Xn=0x-35n=Xn=0(-1)n5n+1(x-3)n,for all|x-3|<1.J. Find Taylor series forf(x) =x2sin(2x)atx= 0Solution:f(x) =x2sin(2x) =x2(2x)-(2x)33!+(2x)55!+......=x2Xn=0(-1)n(2x)2n+1(2n+ 1)!=Xn=0(-1)n22n+1(2n+ 1)!x2n+3,xR.152
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