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Monday, August 2, 2021

Taylor's theorem

Arcsin expansion

Series related to the power series of arcsin(x)


We show the proof of the following result. The proof relies on the series expansion of arcsin. n=1(2n2)!22n(n!)2=1ln(2)
Proof

Lets start with the expansion series of arcsin(x): n=0(2n)!22n(n!)2(2n+1)x2n+1=arcsin(x)|x|1
Taking the derivative n=0(2n)!22n(n!)2x2n=ddxarcsin(x)=11x2
Multiplying by 1x2 and subtracting 1x2 n=1(2n)!22n(n!)2x2n2=1x21x21x2
Integrating x0n=1(2n)!22n(n!)2x2n2dx=n=1(2n2)!(2n)22n(n!)2x2n1=x0(1x21x21x2)dx
Hence x0(1x21x21x2)dx=[1x2x+1x]x0=1x2x+1x
Therefore n=1(2n2)!(2n)22n(n!)2x2n1=1x2x+1x
Integrating one more time from 0 to 1: 10n=1(2n2)!(2n)22n(n!)2x2n1dx=n=1(2n2)!22n(n!)2=10(1x2x+1x)dx
If x=sin(w) and dx=cos(w)dw 10(1x2x+1x)dx=π20[cot(w)cot(w)cos(w)]dw
Using the following identities cos(w)cot(w)=csc(w)sin(w),cot(w)csc(x)=tan(w2)andsin(w)=2sin(w2)cos(w2)
n=1(2n2)!22n(n!)2=π20[cot(w)cot(w)cos(w)]dw=π20[cot(w)csc(w)+sin(w)]dw=π20[cot(w)csc(w)+sin(w)]dw=π20[tan(w2)+2sin(w2)cos(w2)]dw=[2ln(cos(x2))2cos2(x2)]π20=1ln(2)
Therefore n=1(2n2)!22n(n!)2=1ln(2)

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