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If we have two convergent series of positive reals, $∑b_n$ and $∑c_n$, can we find a third convergent series of positive reals, $∑a_n$ , such that $\frac{a_n}{b_n }$ $\rightarrow$ $\infty$ and $\frac{a_n}{c_n }$ $\rightarrow$ $\infty$ ?

I know this can be done with specific series, but can it be done with arbitrary $∑b_n$ and $∑c_n$ ?

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    $\begingroup$ Make $u_n/b_n\to\infty$, make $v_n/c_n\to\infty$, let $a_n=\max(u_n,v_n)$. $\endgroup$
    – Gerry Myerson
    Commented Feb 17, 2022 at 5:17
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    $\begingroup$ Would $u_n$ and $v_n$ in this solution be sequences of positive reals, thus making $a_n$ also a sequence of positive reals? Using this solution, it seems it could be extended to find a suitable $a_n$ for any number of series we are given. $\endgroup$
    – Michael B
    Commented Feb 17, 2022 at 5:30
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    $\begingroup$ @GerryMyerson Indeed, this argument handles any finite number of series, but combining it with a diagonalization we should be able to handle any countable infinity of series. $\endgroup$
    – Andreas Blass
    Commented Feb 17, 2022 at 6:16
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    $\begingroup$ On those matters, I find almost always invaluable have a look at the wonderful book by Konrad Knopp, Theory and application of infinite series, Transl. from the 2nd ed. and revised in accordance with the fourth by R. C. H. Young. (English) London-Glasgow: Blackie & Son, Ltd. XII, 563 p. (1951), Zbl 0042.29203. $\endgroup$
    – Daniele Tampieri
    Commented Feb 17, 2022 at 16:57
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    $\begingroup$ @DanieleTampieri thank you so much for the book recommendation, I'll be sure to check this out. $\endgroup$
    – Michael B
    Commented Feb 18, 2022 at 7:18

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