If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \infty$ then $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} f_{n}(k) = \infty$.

Question

Suppose $\{f_{n}\}_{n=1}^{\infty}$ be functions such that $f_{n} : \Bbb{N} \rightarrow \Bbb{R}^{+}$ for each $n$.

I was trying to prove -

If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \infty$ then $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} f_{n}(k) = \infty$.

I can see that both the double summation series are equal by expanding the double summation.But I am trying to prove it ? any other thoughts?

Hm, as the order of summation are changed, is Uniform convergence likely to play any role here?

Answer 1

With nonnegative summands we have regardless of convergence of the inner sum to a finite value or $\infty$,

$$\sum_{n=1}^N \sum_{k=1}^\infty f_{nk} = \sum_{k=1}^\infty \sum_{n=1}^N f_{nk}$$ and by the MCT

$$\sum_{n=1}^\infty \sum_{k=1}^\infty f_{nk} = \sum_{k=1}^\infty \sum_{n=1}^\infty f_{nk}$$