How to derive the gregory series for inverse tangent function? How to derive the Gregory series for the inverse tangent function?
Why is Gregory series applicable only to the set $ [-\pi/4,\pi/4] $ ?

I think the most straightforward way is to yes the expansions of log(1+x) and log(1-x)
log(1+x)=x - x^2/2 + x^3/3....and so on.
log(1-x)=-x-x^2/2-x^3/3....and so on.
Now subtract the two series to get
log[(1+x)/(1-x)]=
2[x+x^3/3+x^5/5+x^7/7......inf ]
Now replace x with [¡.arctanx] 
After a little bit of manipulation [using  Euler's form]
You will get the famous 
Madhav-Gregory Series .
