# Absolute value convergence vs. convergence (Are they the same?)

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• June 27th 2011, 11:42 PM
CountingPenguins
Absolute value convergence vs. convergence (Are they the same?)
The question is if the absolute value of a sequence converges then does the sequence itself converge. My gut reaction to this is no because you could have something with opposite signs (+ - + -) that would converge in absolute value but diverge otherwise, or even converge to different values. I know that lim n-> infinity |n/-n|=1 and n/-n =-1 but they both converges. No luck on the convergent, divergent scenario. (Keep in mind this is just some random idea about this problem from my head). What do you think?
• June 27th 2011, 11:46 PM
girdav
Re: Absolute value convergence vs. convergence (Are they the same?)
That's the idea indeed. Maybe you could take an explicit example $u_n:=(-1)^n$.
• June 28th 2011, 12:01 AM
CountingPenguins
Re: Absolute value convergence vs. convergence (Are they the same?)
Thanks. I know that putting |1/(-1)^n| converges to 1 and without the absolute value it doesn't. Thanks again.
• June 28th 2011, 02:03 AM
Prove It
Re: Absolute value convergence vs. convergence (Are they the same?)
Quote:

Originally Posted by CountingPenguins
The question is if the absolute value of a sequence converges then does the sequence itself converge. My gut reaction to this is no because you could have something with opposite signs (+ - + -) that would converge in absolute value but diverge otherwise, or even converge to different values. I know that lim n-> infinity |n/-n|=1 and n/-n =-1 but they both converges. No luck on the convergent, divergent scenario. (Keep in mind this is just some random idea about this problem from my head). What do you think?

A series is said to be convergent if you can show that it satisfies a convergence test. Testing for absolute convergence is merely one of those tests. In other words, if a function is absolutely convergent, it is convergent, however, if a function is convergent it may or may not be absolutely convergent.

A perfect example is the alternating harmonic series. It is convergent and has a value of $\displaystyle \ln{2}$, but it is not absolutely convergent, because the series of absolute values is the harmonic series, which is divergent.
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