/sci/ - Science & Math

when is it true that something like [eqn]\min_{y} f(x,y) g(x,y) = \left[\min_{y} f(x,y) \right] g\!\left(x, \operatorname{argmin}_{y} f(x,y)\right)[/eqn] holds? that is, you can minimize the functions separately.

Let [math]\{a_n\}_n[/math] be a sequence of non-negative numbers such that [math]\sum_{i=0}^\infty a_n < \infty[/math] and [math]\sum_{i=0}^\infty a^2_n < \infty[/math].

Fix some positive constant [math]c[/math]. What is [math] \sum_{i=0}^\infty a_n c^n[/math]?

I know that the series converges since [math]\sum_{i=0}^\infty c^n = 1/(1-c) < \infty[/math], but I can't evaluate the limit for an arbitrary [math]a_n[/math].