A stupid one... (just to be sure)
If I have a series that is converge absolutely so the series is converges?
(it is implies from Cauchy criterion?)
Thanks!
A stupid one... (just to be sure)
If I have a series that is converge absolutely so the series is converges?
(it is implies from Cauchy criterion?)
Thanks!
Actually it is.
Note that for any infinite series $\displaystyle \sum{a}$, it can never be any greater than the sum of its absolute values.
So $\displaystyle \sum{a} \leq \sum{|a|}$.
Now simply apply the comparison test.
Note though that this only half proves the theorem, however you can also bound the series below...
$\displaystyle \sum{(-|a|)} \leq \sum{a}$
$\displaystyle -\sum{|a|} \leq \sum{a}$.
So we can say
$\displaystyle -\sum{|a|} \leq \sum{a} \leq \sum{|a|}$
and since the series is bounded by the sum of the absolute values, that means if the sum of the absolute values converges, then the series is also convergent.