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(2n−2)!22n(n!)2=1−ln(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)=1√1−x2
Multiplying by 1x2 and subtracting −1x2
∞∑n=1(2n)!22n(n!)2x2n−2=1x2√1−x2−1x2
Integrating
∫x0∞∑n=1(2n)!22n(n!)2x2n−2dx=∞∑n=1(2n−2)!(2n)22n(n!)2x2n−1=∫x0(1x2√1−x2−1x2)dx
Hence
∫x0(1x2√1−x2−1x2)dx=[−√1−x2x+1x]x0=−√1−x2x+1x
Therefore
∞∑n=1(2n−2)!(2n)22n(n!)2x2n−1=−√1−x2x+1x
Integrating one more time from 0 to 1:
∫10∞∑n=1(2n−2)!(2n)22n(n!)2x2n−1dx=∞∑n=1(2n−2)!22n(n!)2=∫10(−√1−x2x+1x)dx
If x=sin(w) and dx=cos(w)dw
∫10(−√1−x2x+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(2n−2)!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=1−ln(2)
Therefore
∞∑n=1(2n−2)!22n(n!)2=1−ln(2)
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