I will do another example on my own

Question

idku · Answer

$$f(a)+f'(a)(x-a)^1+\frac{f''(a)}{2!}(x-1)^2+\frac{f'''(a)}{3!}(x-1)^3+~...$$Above is a basic representation (expansion) of a taylor series.

idku · Answer

I will now attempt to write a taylor series for sin(x) at x=0, otherwise known as the maclauren series.

idku · Answer

f(0)=sin(0)=0
f'(0)=cos(0)=1
f''(0)=-sin(0)=0
f'''(0)=-cos(0)=-1

then the cycle of coefficients (i.e.  0, 1, 0, -1 ) repeats.

f''''(0)=sin(0)=0
and on....

idku · Answer

$$\sin(x)~~~({\rm at~~x=0})~~\ ~\ =~x-\frac{x^3}{3!}(x-1)^3+\frac{x^5}{5!}(x-1)^3+~.....$$

idku · Answer

Correction:

oh, the x-1 chould be just x, since it is at x=0.

idku · Answer

$$x-x^3/3!+x^5/5!-x^7/7!+...=\sum_{n=0}^{\infty}(-1)^{n}x^{2n-1}/(2n-1)!$$