Exercise 2.8.2

Question

Show that if the iterated series
    $$ \sum^\infty_{i = 1} \sum^\infty_{j=1} \left| a_{ij} \right| $$
    converges (meaning that for each fixed $i \in \mathbf{N}$ the series $\sum^\infty_{j=1}|a_{ij}|$ converges to some real number $b_i$, and the series $\sum^\infty_{i=1}b_i$ converges as well), then the iterated series
    $$\sum^\infty_{i=1} \sum^\infty_{j=1} a_{ij}$$
    converges.

Ulisse Mini · Accepted Answer

Since $\sum^\infty_{j=1}|a_{ij}|$ converges, $c_i = \sum^\infty_{j=1}a_{ij}$ converges as well; moreover $|c_i| \leq |b_i|$, so $\sum^\infty_{i=1}c_i$ converges by comparison with $\sum^\infty_{i=1}|b_i|$.

Exercise 2.8.2

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Exercise 2.8.2

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