How do you represent f(x+h) and f(x−h) as a Taylor series using the taylor series formula?

Question

I know the answers are below, however i am not quite sure what to substitute as the "a" in the Taylor series formula.

$f(x+h)=f(x)+f′(x)⋅h+\frac 12f′′(x)\cdot h^2+\cdots+\frac 1{n!}f^{(n)}(x) \cdot h^n$

and

$f(x−h)=f(x)−f′(x)⋅h+\frac 12f′′(x)\cdot h^2+\cdots+ \frac 1{n!}f^{(n)}(x) \cdot (-h)^n$

Answer 1

I personally like to write the Taylor series as follows:

$$f(x)=f(a)+f'(a)\Delta x+\frac{f''(a)}{2!}(\Delta x)^{2}+\mathcal{O}(\Delta x)^{3}$$

Where $a$ is the point at which you are expanding about, and $\Delta x = x-a$. This usually helps me when trying to work out expansions around a point.

Here, for your first example, you have $a=x$ and $\Delta x = h$, for the second you have $a=x$ and $\Delta x = -h$. Therefore for your second example we get:

$$f(x-h)=f(x)-f'(x)(-h)+\frac{f''(x)}{2}(-h)^{2}+\mathcal{O}(h)^{3}$$

I hope this helps!!