Proving $\sum^\infty_{n=1}a_n$ converges absolutely iff each each sub series converges

Asked Apr 30 '14 at 08:15

Active Apr 30 '14 at 08:15

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We have a series $\displaystyle\sum^\infty_{n=1}a_n$ and a sub series $\displaystyle\sum^\infty_{k=1}a_{n_k}$ where $n_k\in\mathbb N$.

Prove that $\displaystyle\sum^\infty_{n=1}a_n$ converges absolutely iff each each sub series converges.

Hint: choose two converging sub series and use this to prove the original series converge absolutely.

$\Leftarrow$ Let there be two sub series for the odd and one for the even indexes of $a_n$, if both converge than the parant series must converge (but what if only one of them converge ? isn't that a counter example ?).

$\Rightarrow$ Suppose $\displaystyle\sum^\infty_{n=1}a_n$ does not converge absolutely, than one of it's sub series diverges, in contradiction to the convergence of both sub series.

edited Jun 12 '20 at 10:38

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asked Apr 30 '14 at 08:15

GinKin

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Notice that $\sum a_n$ is a subseries of $\sum a_n$... – Najib Idrissi Apr 30 '14 at 08:16
@NajibIdrissi True, but that only means that $\sum a_n$ converges. OP needs to prove that the sum converges absolutely. – 5xum Apr 30 '14 at 08:17
Right. Well consider the subseries of positive terms and the subseries of negative terms, then. – Najib Idrissi Apr 30 '14 at 08:20
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Take as subseries one with all the positve terms $b_n$, and one with all the negative terms $c_n$. Then the $\sum |a_n|=\sum b_n - \sum c_n$ which will exist. – Pol van Hoften Apr 30 '14 at 08:20
you are assuming each subseries is convergent right ? – Airbag Apr 30 '14 at 10:41
@user5656 well if one of them isn't converging, then unless it was like in the example of user45878, the original won't converge absolutely. – GinKin Apr 30 '14 at 17:11

Proving $\sum^\infty_{n=1}a_n$ converges absolutely iff each each sub series converges

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