Can someone find an example of two divergent series whose double series is convergent? That is to say, $\displaystyle\sum_{m=1}^\infty F(m,n)=\infty~\text{for all}~n\in\mathbb{N}$, and $\displaystyle\sum_{n=1}^\infty F(m,n)=\infty~\text{for all}~m\in\mathbb{N}$, but $\displaystyle\sum_{m=1}^\infty\sum_{n=1}^\infty F(m,n)

Question

Can someone find an example of two divergent series whose double series is convergent? That is to say,  $\displaystyle\sum_{m=1}^\infty F(m,n)=\infty~\text{for all}~n\in\mathbb{N}$, and  $\displaystyle\sum_{n=1}^\infty F(m,n)=\infty~\text{for all}~m\in\mathbb{N}$, but  $\displaystyle\sum_{m=1}^\infty\sum_{n=1}^\infty F(m,n)<\infty$.

anonymous · Answer

$$\large\sum_{m=1}^\infty F(m,n)=\infty~\text{for all}~n\in\mathbb{N}$$$$\large\sum_{n=1}^\infty F(m,n)=\infty~\text{for all}~m\in\mathbb{N}$$$$\large\sum_{m=1}^\infty\sum_{n=1}^\infty F(m,n)<\infty$$I was thinking something along the lines of the classic "Gabriel's horn paradox", where the area under ${1\over x}$ from  $x=1$ to $\infty$ is infinite, but the volume contained in the solid of revolution over the same interval is finite.

anonymous · Answer

That's not a very good example for this question though... :|

Here, I'll draw a picture!

anonymous · Answer

|dw:1346390103600:dw|