In general, is $\sum_i \sum_j f(i,j) = \sum_j\sum_i f(i,j)$ ?
With $f(i,j)$ I mean some expression that depends on $i$ and $j$.
If yes, how could I prove that?
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1This holds for FINITE sums. As for the proof: isn't it obvious? Addition is commutative. – Crostul Jun 04 '15 at 10:53
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@Crostul sure. What are the infinite cases? – Simon S Jun 04 '15 at 10:54
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Maybe the OP should be more specific, since its summations are dependent on some unspecified set of indices. – Crostul Jun 04 '15 at 10:55
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@SimonS Infinite cases are those that add up to infinity; these series are typically found in taylor series and such – Conor O'Brien Jun 04 '15 at 10:55
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By infinite cases I mean when the index sets for $i$ and $j$ are (countably) infinite. What's interesting then is when the sums do commute. I'm hesitating in giving an answer because I think I know a very strong sufficient condition, but I'd rather have someone chime in who works with these sorts of expressions more than I do. – Simon S Jun 04 '15 at 11:02
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If I recall correctly, this is a special case of Tonelli's Theorem and is valid iff $f(i,j)$ is non-negative $\forall~i,j$. – Prasun Biswas Jun 04 '15 at 11:08
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I (OP) do want to include infinite cases – Christian Jun 04 '15 at 11:17
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That's the very strong sufficient condition I was thinking of. The "only if" direction of that implication is false, e.g., if there are finitely many negative terms. Now, what if there are infinitely many negative terms? – Simon S Jun 04 '15 at 11:17
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possible duplicate of Positivity and Interchange of Summation – Simon S Jun 04 '15 at 16:26
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$$\sum_i\sum_jf(i,j) = \sum_i( f(i,j_1) + f(i,j_2)+...+f(i,j_n)) =$$$$ \sum_i f(i,j_1)+\sum_if(i,j_2)+...+\sum_i f(i,j_n)=$$ $$[f(i_1,j_1)+f(i_1,j_2)+...+f(i_1,j_n)]+[f(i_2,j_1)+f(i_2,j_2)+...+f(i_2,j_n)]+...+[f(i_m,j_1)+f(i_m,j_2)+...+f(i_m,j_n)] =$$ $$ [f(i_1,j_1)+f(i_2,j_1)+...+f(i_m,j_1)]+[f(i_1,j_2)+f(i_2,j_2)+...+f(i_m,j_2)]+...+[f(i_1,j_n)+f(i_2,j_n)+...+f(i_m,j_n)] =$$ $$ \sum_j f(i_1,j)+\sum_j f(i_2,j)+...+\sum_j f(i_m,j) = \sum_j\sum_if(i,j).\square$$
Rayhunter
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I guess there is a misprint in the derivation above in the line "3" where you expanded in "j" not in "i". – Konstantin Burlachenko Sep 16 '21 at 16:25