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What does it mean by a series of real number $\sum_{i=1}^{\infty}a_n$ converges absolutely to a real number $a$?

a) $\sum_{i=1}^{\infty}|a_n| = a$

b) $\sum_{i=1}^{\infty}|a_n|$ converges and $\sum_{i=1}^{\infty}a_n = a$

For example, there was a theorem on Cauchy product that used the notation of series converges absolutely to a real number.

If $\sum_{n=1}^{\infty}a_n$ converges absolutely to $a$ and $\sum_{n=1}^{\infty}b_n$ converges absolutely to $b$, then $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_i b_j = ab$

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  • $\begingroup$ Notice is a sequence of numbers can jump back end forth from negative to positive. So a series can converge even if the sum of the absolute vales do not. Example, the harmonic series 1+1/2+1/3+1/4 diverges. But 1-1/2+1/3-1/4 obvious can't diverge as every time it goes down or up, the next step is not large enough to get back to where it was before. $\endgroup$
    – fleablood
    Apr 21 '17 at 2:43
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It's b). It's a shortened combination of "$\sum_{i=1}^{\infty} a_n$ converges absolutely" and "$\sum_{i=1}^{\infty} a_n$ converges to $a$".

a) would be something like "the (infinite) sum of the absolute values converges to $a$" (but actually this is so rarely seen that I've come up with that ad hoc, rather than from a standard set phrase).

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