Can you please explain me what does the remainder of a taylor series means?
Also,
How to expand sin x to infinity for all x,by validity power series expansion?
Can you please explain me what does the remainder of a taylor series means?
Also,
How to expand sin x to infinity for all x,by validity power series expansion?
A Taylor series is a (usually) infinite series representation of a function in an interval. So if you sum only a finite number of terms of the series you will be left with an error between the sum to infinity and the finite sum, this error is what the remainder represents.
RonL
What is the difficulty that you are having, you presumably have the definition of a Taylor or McLaurin series for a function, in which case you just need to know the derivatives of $\displaystyle \sin$:
$\displaystyle \frac{d}{dx} \sin(x)=\cos(x)$
$\displaystyle
\frac{d^2}{dx^2} \sin(x)=\frac{d}{dx}\cos(x)=-\sin(x)
$
so in general:
$\displaystyle
\frac{d^{2n}}{dx^{2n}} \sin(x)=(-1)^n \sin(x)
$
and:
$\displaystyle \frac{d^{2n+1}}{dx^{2n+1}} \sin(x)=(-1)^n \cos(x)$
RonL