Question

I will do another example on my own

Answer 1

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

Answer 2

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

Answer 3

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....

Answer 4

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

Answer 5

Correction:

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

Answer 6

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