What does it mean by taylor series of a function about a point?

Question

I understand what a taylor series mean and how does it work but I don't understand what it means for it to be about a point or does that point have any outcome on the result of the series.

Answer 1

If the series converges, it will just converge to the value of that function (it does not depend on $a$).

When you have a Taylor series around a point $a$, it will mean that if you take a partial sum of the series, it will approximate the function well in an environment of that point $a$ but that approximation will get worse as you get further away from it.

The choice of your point $a$ also can have an effect on the interval your Taylor series converges on.

Answer 2

The Taylor series for $f(x)$ about the point $x = x_0$ is $\sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(x_0) (x - x_0)^n$, where $f^{(n)}(x_0)$ is the $n$th derivative of $f$, evaluated at $x_0$.

The Maclaurin series for $f(x)$ is the Taylor series for $f$ about the point $x = 0$, so it reduces to $\sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) x^n$.

As a slightly trivial example, the Taylor series of the function $f(x) = x^2$ about $x = 0$ is just $x^2$, but about $x = 1$ it is $1 + 2(x - 1) + (x - 1)^2$.

The choice of which point to centre the Taylor series at can have an impact on how it converges, but it's also relevant if you're planning on cutting the Taylor series off to approximate the function with a polynomial, since you typically choose a point that's in the middle of the region you're most interested in.