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I know that if a series is absolutely convergent the terms can be rearranged and the new series would still converge to the same value.

Suppose $A=\sum_n^\infty a_nb_n$ which is absolutely convergent and for each $n$, $b_n=\sum_k^\infty c^n_k$ and each $b_n$ is absolutely convergent.

Now I know that for each $b_n=\sum_k^\infty c^{n}_k$ the $c^{n}_k$ can be rearranged and the sum is the same. Similarly, for A the $a_nb_n$ can be rearranged and the sum is still the same.

However, my prof said (in the proof of some theorem), that $A=\sum_n\sum_k a_nc^{n}_k=\sum_k\sum_n a_nc^{n}_k$ where the switch in order of summation is allowed by absolute convergence.

This is what I do not understand. I think the switch in order of summation involves a different rearrangement than just rearranging just the $a_nb_n$ or the $c^{n}_k$ in each $b_n$ because now you have terms in $b_j$ summing with terms in $b_i$.

Can someone please explain why the switch is allowed?

Jhon Doe
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    You might want to search for "Fubini's theorem for series". I tried this and got the following link https://math.stackexchange.com/questions/2063147/proof-of-fubinis-theorem-for-infinite-sums?rq=1 – Calvin Khor Oct 16 '19 at 05:46
  • found this link https://math.stackexchange.com/questions/504445/zeta-question-prime-zeta-basic-calculus/504458#504458 – Jhon Doe Oct 16 '19 at 08:15
  • That is the equality for non-negative terms. Notably, this is assumed (or at least assumed to be easy ;) ) in the link I found – Calvin Khor Oct 16 '19 at 08:17
  • Ah right the terms here might not be non-negative – Jhon Doe Oct 16 '19 at 08:18
  • Another name that you probably already saw from searching, the equality for non-negative terms is sometimes called Tonelli's Theorem (And Fubini's theorem, or the two together, are sometimes called the Fubini-Tonelli Theorem (for series)) – Calvin Khor Oct 16 '19 at 08:19

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