Arctangent series
In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function: arctan x = x − x 3 3 + x 5 5 − x 7 7 + ⋯ = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 2 k + 1 . {\displaystyle \arctan x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{2k+1}}.} This series converges in the complex disk | x | ≤ 1 , {\displaystyle |x|\leq 1,} except for x = ± i {\displaystyle x=\pm i} (where arctan ± i = ∞ {\displaystyle \arctan \pm i=\infty } ).