Is there a formula for the square of an infinite sum?

[Char. Limit]

Given a general, convergent infinite sum, say, one like this:

\sum_{n=0}^\infty a_n x^n

Is there a formula for the square of such a sum? I looked at the Cauchy product, but I'd rather not get a double sum as an answer...

 

[LCKurtz]

Given a general, convergent infinite sum, say, one like this:

\sum_{n=0}^\infty a_n x^n

Is there a formula for the square of such a sum? I looked at the Cauchy product, but I'd rather not get a double sum as an answer...

YMMV but I wouldn't call the Cauchy product a double sum. And if you wish to write the result as a power series, you're stuck with it. Those finite inner sums give the coefficients of $x^n$. It is what it is.

 

[Char. Limit]

YMMV but I wouldn't call the Cauchy product a double sum. And if you wish to write the result as a power series, you're stuck with it. Those finite inner sums give the coefficients of $x^n$. It is what it is.

Aww, well I guess if I need to use the Cauchy product I'll have to figure out how to use it... thanks for the help though!