Power series with square summable coefficients An exercise in my textbook asks to prove the following:


  Suppose $\sum |a_n|^2&lt;\infty$. Show that the power series $\sum
 a_nz^n$ has radius of convergence at least equal to $1$.


My reasoning is as follows:

Since $\sum |a_n|^2&lt;\infty$ we should have $|a_n|^2\to 0$ and hence $|a_n|\to 0$.

Then $\limsup |a_n|^{1/n}\le 1$ and hence the radius of convergence is at least one.

Although I believe my solution is correct, I still don't understand the reason for assuming square summability. We could have assumed that $\sum |a_n|^r&lt;\infty$ for any $r&gt;0$, couldn't we? Does $r=2$ have any particular application or significance?
Your arguments are correct and you are correct that it would work for any $r &gt; 0$. The main point is probably that square-summability is a weaker condition than (absolute) summability, yet still sufficient to guarantee the result. It might not be quite as obvious without Hadamard's formula.
According to your  radius of convergence is at most one no