Convergence of power series

Some thoughts on the convergence of power series - with pictures.

Consider the three functions

f₁(x) = 1 / (1 - x²)

f₂(x) = 1 / (1 + x²)

f₃(x) = √(1 + x²)

If we expand each of these in a power series around x = 0 we find that

f₁(x) = Σ x²ⁿ

f₂(x) = Σ (-1)ⁿx²ⁿ

f₃(x) = Σ x²ⁿ Binomial(1/2,n)

and that each converges for -1 < x < 1.

In the case of f₁, it's easy to see why that's the case. Here's the graph of f₁, and it behaves badly at -1 and 1.

In the case of f₂, it's not immediately obvious why reaching -1 and 1 causes problems. Here's the graph of f₂, and it certainly doesn't behave badly at all at these points.

On the other hand, the graph of what we get by summing the first few terms in the power series for f₂ looks like this, which does behave badly at -1 and 1.

This is easy to explain. If we look at |f₂| as a function of a complex variable, we see that f₂ has poles at i and –i, and the location of those poles limits the radius of convergence of the power series.

In the case of f₃, it's slightly more complicated. Here's what f₃ looks like. It also doesn't behave badly at -1 or 1.

Here's what |f₃| looks like as a function of a complex variable, and there aren't any poles there to cause problems. What's going on here?

What's causing problems in this case are the branch cuts that you need to define for f₃. Here's a graph of the imaginary part of f₃. Note that it's the locations of the branch cuts that limit the radius of convergence of the power series.

I don't recall being taught about branch cuts limiting the radius of convergence of a power series when I was in school. Like many other things, I'm not sure why this was overlooked.