What is the first three non-zero terms in the Taylor series expansion for arctan[e^(−x)−1]

Question

$$f(x) = \arctan[e^{-x} - 1],~~~~~~~~ a = 0$$

$~a~$ is a specified point

I tried doing differentiation, but it is just too tedious.

Answer 1

The first terms of the Taylor series of $\arctan$ at $0$ are $x-\frac{x^3}3$ and the first terms of the Taylor series at $0$ of $e^{-x}-1$ are $-x+\frac{x^2}2-\frac{x^3}3$ (here, “first terms” means the terms with degree up to $3$).

So, compute$$-x+\frac{x^2}2-\frac{x^3}3-\frac13\left(-x+\frac{x^2}2-\frac{x^3}3\right)^3$$and then forget the terms with degree greater than $3$. You will get$$-x+\frac{x^2}2+\frac{x^3}6,$$which is the answer to your question.