## Reversal of Indices in Infinite Triangular Sums

We have:

$\sum_{i=0}^{\infty} \sum_{j=i}^{\infty} \varphi(i,j) = \sum_{j=0}^{\infty} \sum_{i=0}^{j} \varphi(i,j)$

because

$\sum_{i=0}^{\infty} \sum_{j=i}^{\infty} \varphi(i,j) = \sum_{\begin{array}{c}i,j \\ 0 \le i \le j < \infty\end{array}} \varphi(i,j) = \sum_{j=0}^{\infty} \sum_{i=0}^{j} \varphi(i,j)$

In an earlier post, I showed the finite version of this result:

$\sum_{i=0}^{n} \sum_{j=i}^{n} \varphi(i,j) = \sum_{j=0}^{n} \sum_{i=0}^{j} \varphi(i,j)$

This might be seen to hold as both sums are

$\sum_{\begin{array}{c}i,j \\ 0 \le i \le j \le n\end{array}} \varphi(i,j)$