Can anyone recommend some book that discusses the convergence of double series?
Is that following relation holds under certain conditions?
sum_{k,l=1}^{infinity}(a_k+b_l)=sum_{k=1}^{infinity}(a_k)+sum_{l=1}^{infinity}(b_l)
A good reference for the list of definitions and properties of double series is http://www.encyclopediaofmath.org/index.php/Double_series
The equality you are asking about is not true. Say, if a_1 = 1 and all the remaining a_k and b_l are zeros, then the left hand side is infinity (1 + 1 + 1 + ...) but the right hand side is 1. Moreover, the only case when the left hand side is a convergent double series is when all its terms are zeros.
Dear,Mr Muhammas Ali,
i suggest you this book, hope that helps you,
http://bookos-z1.org/book/441658/e89c1c
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