Does absolute convergence of a sequence imply convergence?
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In my real analysis notes I've got that absolute convergence of a real SERIES implies convergence of the series. However what about absolute convergence of a sequence? Does this imply convergence of the sequence?
real-analysis
asked Aug 13 '14 at 21:53
Spongebob SquarepantsSpongebob Squarepants
373
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add a comment |
$begingroup$
In my real analysis notes I've got that absolute convergence of a real SERIES implies convergence of the series. However what about absolute convergence of a sequence? Does this imply convergence of the sequence?
real-analysis
asked Aug 13 '14 at 21:53
Spongebob SquarepantsSpongebob Squarepants
373
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I have not seen the notion of absolute convergence of a sequence used anywhere.
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– André Nicolas
Aug 13 '14 at 22:55
add a comment |
$begingroup$
In my real analysis notes I've got that absolute convergence of a real SERIES implies convergence of the series. However what about absolute convergence of a sequence? Does this imply convergence of the sequence?
real-analysis
asked Aug 13 '14 at 21:53
Spongebob SquarepantsSpongebob Squarepants
373
$endgroup$
In my real analysis notes I've got that absolute convergence of a real SERIES implies convergence of the series. However what about absolute convergence of a sequence? Does this imply convergence of the sequence?
real-analysis
real-analysis
asked Aug 13 '14 at 21:53
Spongebob SquarepantsSpongebob Squarepants
373
asked Aug 13 '14 at 21:53
Spongebob SquarepantsSpongebob Squarepants
373
asked Aug 13 '14 at 21:53
Spongebob SquarepantsSpongebob Squarepants
373
asked Aug 13 '14 at 21:53
Spongebob SquarepantsSpongebob Squarepants
373
asked Aug 13 '14 at 21:53
Spongebob SquarepantsSpongebob Squarepants
373
373
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I have not seen the notion of absolute convergence of a sequence used anywhere.
$endgroup$
– André Nicolas
Aug 13 '14 at 22:55
add a comment |
$begingroup$
I have not seen the notion of absolute convergence of a sequence used anywhere.
$endgroup$
– André Nicolas
Aug 13 '14 at 22:55
$begingroup$
I have not seen the notion of absolute convergence of a sequence used anywhere.
$endgroup$
– André Nicolas
Aug 13 '14 at 22:55
$begingroup$
I have not seen the notion of absolute convergence of a sequence used anywhere.
$endgroup$
– André Nicolas
Aug 13 '14 at 22:55
add a comment |
1 Answer
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No. e.g. $(-1)^n$ does not converge but $left|(-1)^nright|=left|-1right|^n=1 quad $ does converge (to $1$).
As NotNotLogical has pointed out, the exception to this is when a sequence connverges absolutely to $0$, in which case the sequence converges to $0.$
answered Aug 13 '14 at 21:54
beep-boopbeep-boop
7,97452860
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Worth adding that sequences which absolutely converge to zero will also converge to zero, but that is the only exception.
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– user142299
Aug 13 '14 at 21:55
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@NotNotLogical Well noted. Thanks!
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– beep-boop
Aug 13 '14 at 21:57
add a comment |
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1 Answer
1
active
oldest
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1 Answer
1
active
oldest
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active
oldest
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active
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$begingroup$
No. e.g. $(-1)^n$ does not converge but $left|(-1)^nright|=left|-1right|^n=1 quad $ does converge (to $1$).
As NotNotLogical has pointed out, the exception to this is when a sequence connverges absolutely to $0$, in which case the sequence converges to $0.$
answered Aug 13 '14 at 21:54
beep-boopbeep-boop
7,97452860
$endgroup$
$begingroup$
Worth adding that sequences which absolutely converge to zero will also converge to zero, but that is the only exception.
$endgroup$
– user142299
Aug 13 '14 at 21:55
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@NotNotLogical Well noted. Thanks!
$endgroup$
– beep-boop
Aug 13 '14 at 21:57
add a comment |
$begingroup$
No. e.g. $(-1)^n$ does not converge but $left|(-1)^nright|=left|-1right|^n=1 quad $ does converge (to $1$).
As NotNotLogical has pointed out, the exception to this is when a sequence connverges absolutely to $0$, in which case the sequence converges to $0.$
answered Aug 13 '14 at 21:54
beep-boopbeep-boop
7,97452860
$endgroup$
$begingroup$
Worth adding that sequences which absolutely converge to zero will also converge to zero, but that is the only exception.
$endgroup$
– user142299
Aug 13 '14 at 21:55
$begingroup$
@NotNotLogical Well noted. Thanks!
$endgroup$
– beep-boop
Aug 13 '14 at 21:57
add a comment |
$begingroup$
No. e.g. $(-1)^n$ does not converge but $left|(-1)^nright|=left|-1right|^n=1 quad $ does converge (to $1$).
As NotNotLogical has pointed out, the exception to this is when a sequence connverges absolutely to $0$, in which case the sequence converges to $0.$
answered Aug 13 '14 at 21:54
beep-boopbeep-boop
7,97452860
$endgroup$
No. e.g. $(-1)^n$ does not converge but $left|(-1)^nright|=left|-1right|^n=1 quad $ does converge (to $1$).
As NotNotLogical has pointed out, the exception to this is when a sequence connverges absolutely to $0$, in which case the sequence converges to $0.$
answered Aug 13 '14 at 21:54
beep-boopbeep-boop
7,97452860
edited Aug 13 '14 at 22:50
answered Aug 13 '14 at 21:54
beep-boopbeep-boop
7,97452860
answered Aug 13 '14 at 21:54
beep-boopbeep-boop
7,97452860
answered Aug 13 '14 at 21:54
beep-boopbeep-boop
7,97452860
7,97452860
$begingroup$
Worth adding that sequences which absolutely converge to zero will also converge to zero, but that is the only exception.
$endgroup$
– user142299
Aug 13 '14 at 21:55
$begingroup$
@NotNotLogical Well noted. Thanks!
$endgroup$
– beep-boop
Aug 13 '14 at 21:57
add a comment |
$begingroup$
Worth adding that sequences which absolutely converge to zero will also converge to zero, but that is the only exception.
$endgroup$
– user142299
Aug 13 '14 at 21:55
$begingroup$
@NotNotLogical Well noted. Thanks!
$endgroup$
– beep-boop
Aug 13 '14 at 21:57
$begingroup$
Worth adding that sequences which absolutely converge to zero will also converge to zero, but that is the only exception.
$endgroup$
– user142299
Aug 13 '14 at 21:55
$begingroup$
Worth adding that sequences which absolutely converge to zero will also converge to zero, but that is the only exception.
$endgroup$
– user142299
Aug 13 '14 at 21:55
$begingroup$
@NotNotLogical Well noted. Thanks!
$endgroup$
– beep-boop
Aug 13 '14 at 21:57
$begingroup$
@NotNotLogical Well noted. Thanks!
$endgroup$
– beep-boop
Aug 13 '14 at 21:57
add a comment |
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$begingroup$
I have not seen the notion of absolute convergence of a sequence used anywhere.
$endgroup$
– André Nicolas
Aug 13 '14 at 22:55