The double series $\sum u_{\mu, \nu}=\sum\frac{\mu-\nu}{\mu+\nu}$ is said to have sum by rows equaling -1, sum by columns equaling 1. But if we sum by rows, for each row, we have $\sum_{\nu=0}^\infty u_{\mu, \nu}=\sum_{\nu=0}^\infty\frac{\mu-\nu}{\mu+\nu}\approx\sum_{\nu=0}^\infty -1=-\infty$, the sum by row therefore is $\sum_{\mu=0}^\infty\sum_{\nu=0}^\infty u_{\mu, \nu}=\sum_{\mu=0}^\infty-\infty=-\infty$, contradicting the statement at the beginning. Is my calculation wrong?
I made a mistake about the series, its sum $S_{\mu, \nu}=\frac{\mu-\nu}{\mu+\nu}$,
so the sum of each row seems to be $S_{1, \infty}=-1, \sum_{\nu=0}^\infty u_{2, \nu}=\sum_{\nu=0}^\infty u_{3, \nu}=\dots=0$, sum of each column seems to be $S_{\infty,1}=11, \sum_{\mu=0}^\infty u_{\mu, 2}=\sum_{\nu=0}^\infty u_{\mu, 3}=\dots=0$.
It seems a weird behavior for a double series, how is $u_{\mu, \nu}$ like?