<s>[INST] Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are positive and decreasing.
Edit: This problem is taken verbatim from Exercise 2.7.11 on page 68 of Abbott's Understanding Analysis.
 [/INST] This answer gives a specific example using the ideas proposed in the previous answer:
Let $\sum a_n=\overline{\frac{1}{1^2}}+\frac{1}{2^2}+\overline{\frac{1}{3^2}+\frac{1}{3^2}+\frac{1}{3^2}}+\frac{1}{6^2}+\frac{1}{7^2}+\cdots+\frac{1}{14^2}+\overline{\frac{1}{15^2}+\frac{1}{15^2}+\cdots+\frac{1}{15^2}}+\cdots$
and $\sum b_n=\frac{1}{1^2}+\underline{\frac{1}{2^2}}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\underline{\frac{1}{6^2}+\frac{1}{6^2}+\cdots+\frac{1}{6^2}}+\frac{1}{15^2}+\frac{1}{16^2}+\cdots+\frac{1}{71^2}+\cdots$.
Notice that each group of repeated terms has a sum $S\ge\frac{1}{4}$, and that $\sum c_n=\sum\frac{1}{n^2}$.
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