Convergence of Infinite Sums and Limits: A L'Hopital's Rule Perspective

[Apteronotus]

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

 

[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.

 

[Apteronotus]

Thank you L'Hopital!