Are $f(x)$ and $f(x+ \delta x)$ the same after Taylor series expansion? According to  15.2.1 from https://www.rsmas.miami.edu/users/miskandarani/Courses/MSC321/lectfiniteDifference.pdf, the Taylor series of u(x) can be written as

However, according to wikipedia, the Taylor series is 

The difference is in $\delta x$. My question is are $f(x_i)$ and $f(x_i+\delta_x)$ the same?

In the second formula:
$f(x)_{about\space x=a}= f(a) + \dfrac{f'(a)}{1!}(x-a) + \dfrac{f''(a)}{2!}(x-a)^2+\dfrac{f'''(a)}{3!}(x-a)^3$
Replace $x \rightarrow x+\Delta x$ & $a \rightarrow x$ to get:
$f(x+\Delta x)_{about\space x=x}= f(x) + \dfrac{f'(x)}{1!}(\Delta x) + \dfrac{f''(x)}{2!}(\Delta x)^2+\dfrac{f'''(x)}{3!}(\Delta x)^3$
which is the first formula itself.
