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[tex]\hbox {Is }\;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j} \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}B_{i,j}\;\hbox{?}[/tex]

[tex]\hbox {Is }\;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}\;\hbox{?}[/tex]

I think it is because even though the right side has two summation of ##i## , but both increment at the same time. So is ##j##. therefore the result is the same.

[tex]\hbox {Is }\;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \;\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}A_{i,j}B_{i,j}=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} A_{i,j}B_{i,j}\;\hbox{?}[/tex]

I think it is because even though the right side has two summation of ##i## , but both increment at the same time. So is ##j##. therefore the result is the same.

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