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If $\sum a_k$ converges absolutely, then $|a_k|<\frac{1}{k}$ for all sufficiently large k.

I'm trying to give a proof or a counterexample about the above statement, but I'm not really sure where to start. I think it's false, but don't know how to go about finding a counterexample. Any help would be much appreciated.

Thanks

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    Do you mean $|a_k| < \frac {1}{k}$?2012-12-30
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    Yep I did, thanks2012-12-30

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HINT: The idea is to have infinitely many terms $a_k$ such that $a_k\ge\frac1k$, but to spread them out very thinly. Try letting

$$a_k=\begin{cases} \frac1k,&\text{if }k\text{ is a perfect square}\\\\ 0,&\text{otherwise}\;. \end{cases}$$

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    @Jonas: It means that I didn’t include a proof that the example works. I should perhaps have said BIG HINT.2012-12-30
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    I still don't understand how you came up with the example? And why do you want to spread them out thinly?2012-12-30
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    @user51897 The intuition is that $\int \frac {1}{k} dk = \ln k$, so when evaluated from $n=k$ to infinity, has the value $\infty$. This means that the terms need not converge. As such, we need it to be less frequent than that. Brian's intuition is that $\int \frac {1}{k^2} dk = \frac {1}{k}$, which when evaluated as a definite integral, is finite.2012-12-30
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    @user51897: Because the harmonic series diverges, so you can't have "too many" terms such that $a_k\geq \frac{1}{k}$. If you are going to have infinitely many terms such that $a_k\geq \frac1{k}$, they have to be chosen carefully from a set such that those reciprocals *do* converge. The sum of the reciprocals of the squares converges. Another example would be the sum of the reciprocals of the powers of $2$.2012-12-30
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    Ok, so I think I finally understand that. How do you start a proof about an absolutely convergent series?2012-12-30
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    @user51897: That depends very much on what I want to prove. Here the absolute convergence wasn’t really very important. In general the nice thing about absolutely convergent series is that they can be rearranged however you please without affecting the limit.2012-12-30
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    Are absolutely convergent and unconditionally convergent the same thing?2012-12-31
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    @user51897: Absolute convergence is a special case of [unconditional convergence](http://en.wikipedia.org/wiki/Unconditional_convergence).2012-12-31