Taylor formula for y=tanx at x= 0

Question

I need to write a Taylor formula for $y=\tan x$ at the point $x=0$ till the member of second order ( does that means till second derivative or just an element on Taylor expansion?)

so my soultion: $f(0)=0 ; f'(0)=1 ; f''(0)= \frac{2\cdot \sin x}{\cos^3(x)}=0$?

then the Taylor expansion till second order member ( can somepne tell me, what's the appropriate name for it?) is $P_n(x)= 0 + x + ? $

How can I calculate the last element?

Answer 1

Since you computed correctly $$f(0)=0,\quad f'(0)=1,\quad f''(0)=0$$ the second order Taylor polynomial of $f:=\tan$ at $x=0$ is $$j^2_0\tan(x)=0+1\cdot x+{1\over2}\cdot0 \cdot x^2=x\ .$$

Answer 2

Do synthetic division of

$\begin{array}\\ \tan(x) &=\dfrac{\sin(x)}{\cos(x)}\\ &=\dfrac{x-\frac{x^3}{6}+\frac{x^5}{120}...}{1-\frac{x^2}{2}+\frac{x^4}{24}+...}\\ &=x\dfrac{1-\frac{x^2}{6}+\frac{x^4}{120}+...}{1-\frac{x^2}{2}+\frac{x^4}{24}+...}\\ \end{array} $