# Limit of sum, sum of limit

1. Feb 1, 2010

### Apteronotus

For an infinite sum, is the limit of the sum = sum of the limit?
ie.
$$lim_{x \rightarrow a} \sum_{n=0}^\infty f(x,n)= \sum_{n=0}^\infty lim_{x \rightarrow a}f(x,n)$$

2. Feb 1, 2010

### l'Hôpital

I'm fairly certain that it's true if and only if
$$\sum_{n=0}^{\infty} f(x,n)$$

converges uniformly. In general, however, no.

3. Feb 2, 2010

### Apteronotus

Thank you L'Hopital!!!