# Where to choose the point of expansion for taylor series?

Typically, in taylor series, I see an expansion about $$x=0$$ for some function $$f(x)$$ that we're approximating. I always thought this was just done for simplicity since this causes a lot of $$x's$$ to drop out in the infinite/truncated series. Is this actually the reason or is it more sophisticated?

In general, how do you best determine the point of expansion? If a function's domain is all positive, then I don't think it makes sense to expand about 0.

• At which point, Taylor series expand, function must be infinitely differentiable at that point, so, point must be in the domain! Jul 24 '20 at 5:40
• @Subhajit. Sure but which point in the domain to choose?
– 24n8
Jul 24 '20 at 5:46
• I'd choose a point ($x_0$) such that (a) $f(x_0)$ is easy to compute and (b) is close to the point(s) ($y$) for which I want $f(y)$. Jul 24 '20 at 5:51
• If you are taking about real valued function,then all the points of domain where it is infinitely differentiable! Jul 24 '20 at 5:51
• This essentially depends on why you are using a Taylor expansion.
– user65203
Jul 24 '20 at 6:24

This implies that one should do the Taylor series approximation roughly around a point such that this point lies in the middle of the interval for which one wants to evaluate the approximation. Thus, if you are interested in the values of $$f(x)$$ close to $$x=0$$, do the approximation around $$x=0$$. If, on the other hand, you are interested in the behavior of $$f(x)$$ around $$x=100$$, approximate $$f$$ around this point.
Finally, the reason why you typically mainly see approximations around $$x=0$$ is because many people first make a coordinate transformation such that the point they want to approximate around lies at the origin, and only then do the approximation. In many cases, this (slightly) simplifies the steps.