## On convergence

I recently came across something interesting while reading Mary L Boas text on Mathematical Methods.

Has it ever bothered you, why the power series of a pretty function like converges only for , even though there’s nothing strange which happens at ? For example, if you take , it’s easy to see that there’s a singularity at . But what’s the catch here? What’s wrong with the power series of at ? This apparent conundrum is actually trivial when one looks at it from the complex perspective. Consider the complex function,

Since is infinite when , isn’t analytic in any region containing . Now, there’s a theorem that says that, since is analytic at points other than , it converges **inside** the circle centered at the origin and extending to the singularity at . From this, one can clearly see that when , and the corresponding power series also converges only when .

The lesson here is that looking at things from a complex perspective might give important insights in the real case.

Hold yer horses! What do you mean 1/(1+x^2) ‘converges’? That it has a limit as x -> 1 ? But _of course_ it has a limit, and it is 1/2! Why should you even think of the complex plane? I do not also understand what is the problem with it as x -> 1, why should you _suspect_ anything wrong at all, in the first place?

That, or I need more sleep 🙂

Mohan K.VApril 15, 2007 at 1:48 am

Ok, I meant the power series of 1/(1+x^2). I should be more careful with my wording. I’ve edited it now.

parsevalApril 15, 2007 at 9:50 am

Ah, peace 🙂

Mohan K.VApril 17, 2007 at 11:15 am