Construct the first three nonzero terms and the general term of the Maclaurin series of arctan(x^2)

Start with the geometric series
1/(1 - x) = Σ(n = 0 to ∞) x^n.

Let x = -t^2:
1/(1 + t^2) = Σ(n = 0 to ∞) (-1)^n x^(2n).

Integrate both sides from 0 to t:
arctan t = Σ(n = 0 to ∞) (-1)^n t^(2n+1)/(2n+1).

Let t = x^2:
arctan(x^2) = Σ(n = 0 to ∞) (-1)^n x^(4n+2)/(2n+1).
.................= x^2 - x^4/3 + x^6/5 - ...

I hope this helps!