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Thread: Sequence converence conditions

  1. #1
    Oct 2011

    [Please move to "Number theory"] Sequence converence conditions

    Hi, I need some help with some sequences. I need to solve a convergence problem, i.e. to find the conditions/bounds for an infinite sequence $\displaystyle \{ a_i \}_{i=1}^\infty$ of numbers $\displaystyle a_i \in \mathbb{R}^+$, such that the following two conditions hold:

    $\displaystyle (a) \lim_{N \to \infty} \frac{1}{N^2} \sum_{i=1}^N {a_i} = 0$
    $\displaystyle (b) \lim_{N \to \infty} \frac{1}{N} \sum_{i=1}^N \frac{1}{a_i} = 0.$

    I already know one particular family of sequences, which is $\displaystyle \{ i^\alpha \}_{i=1}^\infty$ for $\displaystyle \alpha \in (0;1)$. I can show that for this family (a) and (b) hold, by bounding the sums by the integrals of their corresponding continuous functions, which I can then compute analytically.

    My ultimate goal is to derive tight bounds for the sequence. I feel that my family of solutions already hints the bounds, which I think are $\displaystyle \alpha = 0$ and $\displaystyle \alpha = 1$ as a lower and upper bound. However, I'm not sure.

    For example, I feel that the lower bound is given by (b), which should be that $\displaystyle a_i$ aproaches to infinity, however slowly, as i approaches infinity. Then, the upper bound can be derived from (a) by observing that a linear increase of $\displaystyle a_i$ makes the limit to be a constant, whereas anything sublinear ensures convergence to zero. I'd like a proof that is mathematically sound, however.

    Any ideas will be greatly appreciated!!

    EDIT: I, very sorry, but despite being careful in which forum to put the question, something must have gone wrong, and the post ended up in this forum. Please, can any moderator move it to "Number theory"? Or should I just delete it from here and repost it there?
    Last edited by ingenious; Oct 4th 2011 at 04:50 AM.
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  2. #2
    Senior Member roninpro's Avatar
    Nov 2009

    Re: [Please move to "Number theory"] Sequence converence conditions

    I don't currently have the time to look at everything or be very careful, but I think that you're on the right track.

    The way I interpret condition (b) is that $\displaystyle \displaystyle \frac{1}{N} \sum_{i=1}^N \frac{1}{a_i}$ is the average over the $\displaystyle 1/a_i$, and you want the average to go to zero. The first thing I considered is that if all of your $\displaystyle a_i$ are positive, the only way to do this is to make all of the terms go to zero. But I don't think that this is quite correct. For example, you can take the sequence 1, 1/3, 1, 1/5, 1/6, 1/7, 1,... (i.e. it is the harmonic series with every power of 2 replaced with a 1). It seems to me that the averages will still go to zero even though the sequence is not convergent. (I haven't checked this, but I think that it seems reasonable, especially if the 1's are very sparse.)

    How strict of a condition are you looking for? It seems that there are many possibilities.
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