Question

How do you find the Maclaurin Series for `tan x`?

Answer 1

The answer is `=x+x^3/3+o(x^3)`

Explanation:

Let `f(x)=tanx`, then `f(0)=0`

`f'(x)=sec^2x=1+tan^2x`, `=>`, `f'(0)=1`

`f''(x)=2secx(1+tan^2x)`, `=>`, `f''(0)=2`

The Maclaurin series expansion is

`tanx=f(0)+xf'(0)+x^2/(2!)f''(x)+x^3/(3!)f'''(x)+...`

`=x+x^3/3+o(x^3)`