Say I want to find the Taylor expansion about the point $x_0=1$ of the function $f(x)=e^{2x}$. Following the standard formula, we get the following first few terms.
$$
f(x) = e^{2} + 2e^{2}(x-1) + \frac{4e^{2}(x-1)^2}{2} + \ldots .
$$
However, I sometimes see my lecturer use this notation for the above Taylor expansion.
$$
f(1+\epsilon ) = e^{2} + 2e^{2}\epsilon + \frac{4e^{2}\epsilon^2}{2} + \ldots .
$$
As I see it, the only difference is the reference point, i.e. in the first formulation the reference point is $0$ and in the second it is $1.$ Which is most correct to use?
Best Answer
Neither is more correct. You can ask for the Taylor series about any point. It might or might not exist. If it exists, it will converge within a radius of the distance to the nearest singularity. Both of your functions are real analytic, meaning the Taylor series at both points exists and equals the function, everywhere.
The formula for the Taylor series centered at the point $a$, is $f(x)=\sum_{n\ge0}f^{(n)}(a)(x-a)^n/n!$.
To take a different example, the inverse of $e^x$, $\ln x$, it winds up making a bit more of a difference where you ask for the Taylor series, since it isn't always defined. One often considers $\ln(1+x)$, to avoid the singularity at $0$.