Suppose the infinite series [tex] \sum a_v [/tex] is NOT absolutely convergent. Suppose it also has an infinite amount of positive and an infinite amount of negative terms.(adsbygoogle = window.adsbygoogle || []).push({});

Say we want to prove it converges by proving the sequence of partial sums, A_n, are Cauchy.

Then we need to prove that for every positive epsilon, [tex] |A_n - A_m| < \epsilon [/tex], for n and m sufficiently large.

Note that [tex] |A_n - A_m| = |a_1 + a_2 + ... + a_n - (a_1 + a_2 + ... + a_m)| = |a_{n+1} + a_{n+2} + ... + a_m| [/tex].

Some of these terms are positive, some negative. Are we allowed to "separate" the positive and negative terms? Like this:

Say P_n is the sequence of positive terms and N_n is the sequence of negative terms in the sum [tex] |a_{n+1} + a_{n+2} + ... + a_m| [/tex]. Then can we write [tex] |a_{n+1} + a_{n+2} + ... + a_m| = |P_1 + P_2 + ... + P_i + N_1 + N_2 + ... + N_j| [/tex] ?

It seems to me like this should be fine, since this is a finite sum. But we are going to have to take n and m larger and larger for epsilon getting smaller and smaller, so I am not sure.

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# Arrangement Of Terms Question

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