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Converging series problem

  1. Feb 22, 2006 #1
    "Suppose [tex]a_1\geq a_2\geq ... \geq a_n > 0[/tex] for all [tex]n[/tex] such that [tex]\sum a_n[/tex] converges. Show that [tex]na_n[/tex] tends to zero."

    The best I can do is prove that a subsequence [tex]na_n[/tex] converges to a number between the sum of the series and 0.
     
    Last edited: Feb 22, 2006
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  3. Feb 22, 2006 #2

    NateTG

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    Have you tried assuming that [itex]na_n[/itex] does not tend to zero and using a proof by contradiction?
     
  4. Feb 22, 2006 #3

    AKG

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    Since the series converges, an converges to zero. If nan doesn't converge to 0, then that means that an converges to zero no faster than 1/n. But if an converges no faster than 1/n, then the series will be no "better" than than the harmonic series, and so it certainly will not converge - contradiction. This is the rough idea you might want to work with, but of course, it's not rigorous yet.
     
    Last edited: Feb 22, 2006
  5. Feb 22, 2006 #4
    I thought about that approach, but the problem is that if it converges to a number less than 1, then I can't say that it the term a_n converges no faster than the harmonic series.
     
  6. Feb 22, 2006 #5

    AKG

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    You can use the same idea. If it converges to some number c, such that 1 > c > 0, then an converges no faster than c/n, which means the series is no better than:

    [tex]\sum \frac{c}{n} = c\sum \frac{1}{n}[/tex]
     
  7. Feb 22, 2006 #6
    I've gotten so far:

    na_n is bounded, therefore has a convergent subsequence.

    IF na_n converges, then it converges to 0.

    All I have to do now is prove that na_n converges, which I am unable to.
     
  8. Feb 22, 2006 #7

    NateTG

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    Let's say you have some [itex]n_1[/itex]
    such that
    [tex]n_1a_{n_1} > \epsilon[/tex]

    Can you give a (non-zero) lower bound on
    [tex]\sum_{i=\frac{n_1}{2}}^{n_1} a_i[/tex]

    Now, if, in addition, you had
    [tex]n_2 > 2 \times n_1[/tex]
    with
    [tex]n_2a_{n_2} > \epsilon[/tex]
    can you give a lower bound on
    [tex]\sum_{i=\frac{n_1}{2}}^{n_2} a_i[/tex]
     
  9. Feb 22, 2006 #8
    The lower bound for the first one would be the difference [tex]n_1a_n_1 - n_1/2 a_{n_1/2}[/tex] assuming n1 is even. I don't see how this helps. Are you trying to show that na_n is Cauchy?
     
  10. Feb 22, 2006 #9

    quasar987

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    For an easy plug & chug way: apply the D'Alembert criterion (aka ratio test) on na_n.
     
  11. Feb 22, 2006 #10
    The ratio test is for the series na_n, not the sequence na_n. The fact is that there exists converging a_n such that [tex]\sum a_n[/tex] converges but [tex]\sum na_n[/tex] does not. 1/n^2, for instance.
     
  12. Feb 22, 2006 #11

    AKG

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    CASE 1:
    Suppose {nan}n in N converges to some finite c > 0. Suppose that for all N > 0, there exists n > N such that nan < c/2. Well then {nan}n in N could not converge to c, countradicting our assumption. So there is some N > 0 such that for all n > N, nan > c/2. Thus we can choose a subsequence {nan}n > N which has each entry is at least c/2.

    But then

    [tex]\sum _{n \in \mathbb{N}} a_n = \sum _{n \in \mathbb{N}} \frac{na_n}{n} \geq \sum _{n > N} \frac{na_n}{n} \geq \sum _{n > N} \frac{c}{2n} = \frac{c}{2} \sum _{n > N}\frac{1}{n}[/tex]

    But if the sum on the left converges, then it cannot be greater than the sum on the right, because the one on the right clearly diverges. Contradiction.

    CASE 2:
    {nan}n in N converges to some negative number. Impossible, since all the terms are positive. Contradiction.

    CASE 3:
    {nan}n in N diverges. Same argument as in Case 1. Contradiction.

    CASE 4:
    {nan}n in N converges to 0. By process of elimination, the only posibility, thus it must be the case. Q.E.D.
     
    Last edited: Feb 22, 2006
  13. Feb 22, 2006 #12

    NateTG

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    Let me try it this way:
    What is the minimum value of a term in the sum.
    How many terms are there?
     
  14. Feb 22, 2006 #13
    If it converges to any number, then I can prove that number must be 0:

    [tex]na_n \rightarrow c > 0 \Rightarrow \frac{na_n}{n\frac{1}{n} c}=\frac{a_n}{\frac{c}{n}} \rightarrow 1 \Rightarrow \sum a_n, \sum\frac{c}{n}[/tex] converge or diverge simultaneously. Since [tex]\sum\frac{c}{n}[/tex] diverges, so does [tex]\sum a_n[/tex], which is impossible.

    However, [tex]na_n[/tex] need not converge! All we know is that it's bounded, and therefore by Bolzano-Weierstrass theorem, there exists a converging subsequence. However, the above proof, using the converging subsequence, fails, since [tex]\sum\frac{c}{n_k}[/tex] might be convergent, and thus not leading into a contradiction.

    The problem with your proof is that I can't see how you can conclude for all n>N, since it should be a subsequence only.
     
  15. Feb 22, 2006 #14

    NateTG

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    How's this:

    Assume, by contradiction that
    [tex]\lim_{n \rightarrow \infty} na_n \neq 0 [/tex]

    Then there is some [itex]\epsilon > 0 [/itex] such that for any [itex]n \in \mathbb{N}[/itex] there is some [itex]N > n[/itex] such that [tex]Na_N > \epsilon[/tex].

    So, clearly there is an [itex]n_1>3[/itex] so that [itex]n_1a_{n_1}>\epsilon[/itex], and for any [itex]i>1[/itex] there is some [itex]n_i[/itex] such that [itex]n_i>2\times n_i-1[/itex] and [itex]n_ia_{n_i}>\epsilon[/itex].

    Now [itex]\sum_{j=\ciel{\frac{n_i}{2}}}^{n_i} a_n > \frac{\epsilon}{4}[/itex]
    since the sum contains at least [itex]\frac{n_i}{4}[/itex] terms all greater than [itex]\frac{\epsilon}{n_i}[/itex].

    Since [itex]n_i<\frac{n_{i+1}}^{2}[/itex] we have [itex]n_i<\ceil(\frac{n_{i+1}}{2})[/itex]

    Now, we can regoup
    [tex]\sum_{j=1}^{\infty}a_j \geq \sum_{i=0}^{\infty} \sum_{j=\ciel{\frac{n_i}{2}}}^{n_i} a_j > \sum_{i=0}^{\infty} \frac{\epsilon}{4}[/tex]

    But this contradicts the assumptiont that the sum is convergent.

    Hence the limit must be equal to zero.
     
    Last edited: Feb 22, 2006
  16. Feb 22, 2006 #15

    AKG

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    Which is why I gave case 3.
    What do you mean "all we know is that it's bounded." A priori, it might be unbounded, as it could be that an = 1. On the other hand, since we're asked to prove that {nan} converges to 0, we "know" that not only is it bounded, but that it converges to 0.
    No, I started with:

    Suppose that for all N > 0, there exists n > N such that nan < c/2

    which is:

    [tex](\forall N > 0)(\exists n > N)(na_n < c/2)[/tex]

    I derived a contradiction from this, so I may infer its negation, which is:

    [tex](\exists N > 0)(\forall n > N)(na_n \geq c/2)[/tex]

    which is:

    there is some N > 0 such that for all n > N, nan > c/2.
     
  17. Feb 22, 2006 #16
    I know it's bounded because [tex]\sum_{1}^{\infty}a_i \geq \sum_{1}^{n}a_i \geq na_n > 0[/tex] since a_n is decreasing.
     
  18. Feb 23, 2006 #17

    NateTG

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    I don't see how the argument from case 1 applies to case 3 since there is, by definition, no well-defined limit for a divergent sequence. Moreover, there are divergent sequences where there is no suitable [itex]c[/itex].

    For example, consider that if [itex]a_n=\frac{||-1^n+1|+.5^n}{n}[/itex], then [itex]\left{na_n\right}[/itex] is divergent since it has subsequences that converge to 0 and 2 (the odd and even terms respectively). For any finite [itex]c>0[/itex] and any [itex]n \in \mathbb{N}[/itex], there is some [itex]n>N[/itex] so that [itex]na_n<\frac{c}{2}[/itex] since the odd terms converge to zero.

    I
     
  19. Feb 23, 2006 #18

    AKG

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    Sorry, case 1 doesn't exactly apply to case 3. It does to the subcase that the sequence diverges to infinity. In this case, pick any finite c > 0, and the same argument applies, but you'd replace "could not converge to c" in the second sentence with "could not diverge to infinity". However, case 1 doesn't apply to the subcase where the sequence neither converges nor diverges to infinity, but is simply bounded divergent (i.e., it "oscillates"). Note that Treadstone proved that the sequence is bounded, so we don't really need to apply case 1 to the first subcase of case 3 (the subcase where the sequence diverges to infinity), we can just say that this subcase will never happen

    So, what to do when the sequence oscillates? Well, observe that:

    [tex]\lim _{n \to \infty}\frac{(n+1)a_{n+1}}{na_n} = \lim _{n \to \infty}\frac{n+1}{n}\lim _{n \to \infty}\frac{a_{n+1}}{a_n} = \lim _{n \to \infty}\frac{a_{n+1}}{a_n} \leq 1[/tex]

    since {an} converges, which is true since the series converges. At least, I think that's correct. If it is, then I think it can be used to argue that {nan} in fact does not oscilate, thereby eliminating the remaining subcase of case 3.
     
  20. Feb 23, 2006 #19

    NateTG

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    A cleaner approach

    [tex]na_n = \sum_{i=1}^{n} {a_n} \leq 2 \sum_{i=\lfloor \frac{n}{2} \rfloor}^{n}{a_n} \leq \sum_{i=\lfloor \frac{n}{2} \rfloor}^{\infty}{a_i} \leq \sum_{i=\lfloor\frac{n}{2} \rfloor}^{\infty}{a_i}[/tex]

    The [itex]\leq[/itex] are justified because [itex]a_n[/itex] is strictly positive, [itex]a_n[/itex] is monotone decreasing, and [itex]a_n[/itex] is strictly positive respectively.
    Now we have
    [tex] 0 < na_n \leq \sum_{i=\lfloor\frac{n}{2} \rfloor}^{\infty}{a_i}[/tex]

    Since the sequence converges, the limit on the R.H.S. as n goes to infinity must be zero. This bounds [itex]na_n[/itex] above and below with sequences that go to zero, therfore [itex]na_n[/itex] goes to zero.
     
    Last edited: Feb 23, 2006
  21. Feb 23, 2006 #20

    AKG

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    I think you mean:

    [tex]na_n\ =\ \sum_{i=1}^{n} {a_n}\ \leq\ 2\left ( \sum_{i=\lfloor \frac{n}{2} \rfloor}^{n}{a_n}\right )\ \leq\ 2\left ( \sum_{i=\lfloor \frac{n}{2} \rfloor}^{n}{a_i}\right )\ \leq\ 2\left ( \sum_{i=\lfloor\frac{n}{2} \rfloor}^{\infty}{a_i}\right )[/tex]

    Then applying the squeeze theorem (i.e. your argument) to:

    [tex]0\ <\ na_n\ \leq\ 2\left ( \sum_{i=\lfloor\frac{n}{2} \rfloor}^{\infty}{a_i}\right )[/tex]

    we get the desired result. Very nice!
     
    Last edited: Feb 23, 2006
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