\arctan (\frac{x+1}{x-1}) to power series I want to find an expression

Alan Smith 2021-12-26 Answered
arctan(x+1x1) to power series
I want to find an expression for arctan(x+1x1) as a power series, with x0=0, for every x1
My initial thought was to use the known arctan(x)=n=0(1)nx2n+12n+1 but I don't know how to keep going if I replace x with x+1x1
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Expert Answer

zesponderyd
Answered 2021-12-27 Author has 42 answers
Let f(x)=arctan(x+1x1),  |x|<1
It can be easily showed that:
f(x)=11+x2=n=0(1)nx2n
Integrating both sides yields that CR such that:
arctan(1+x1x)+C=n=0(1)nx2n+12n+1
Check for f(0) to conclude C and you're done.
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Annie Levasseur
Answered 2021-12-28 Author has 30 answers
Use the identity arctan(x+1x1)=arctan(x+11x)=(π4+arctan(x))
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karton
Answered 2022-01-08 Author has 368 answers

Since
ddxarctan(x+1x1)=11+x2
you can express the RHS as a power series, and then integrate the result to get the desired series for your original function arctan(x+1x1)

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