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Prove or give a counter example is sum ai and sum bi are convergent series with non-n

  1. Mar 9, 2012 #1
    Hi

    Can someone please help me to
    prove or give a counter example is sum ai and sum bi are convergent series with non-negative terms then sum aibi converges


    I believe that if it doesn't say "non-negative terms" then this wouldn't be true. Am I correct?

    Since each of two non-negative series converges then the series sum ai bi converges also. However I am not sure how to prove this.

    thanks
     
  2. jcsd
  3. Mar 9, 2012 #2

    jbunniii

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    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    Correct. Can you find a counterexample? (Hint: you can find one with [itex]a_i = b_i[/itex].)

    Do you know any inequalities that involve [itex]\sum a_i b_i[/itex]?
     
  4. Mar 9, 2012 #3
    Re: prove or give a counter example is sum ai and sum bi are convergent series with n


    ai= (cos n pie)/squareroot (n)= bi this would work right?

    I'm sorry I am not sure what you mean....i feel like this problem its very simple but for some reason I have such a hard time with it
     
  5. Mar 9, 2012 #4

    jbunniii

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    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    Yes, that's the example I had in mind. Of course [itex]\cos(n\pi) = (-1)^n[/itex].

    Do you know the Cauchy-Schwarz inequality?
     
  6. Mar 9, 2012 #5
    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    the Cauchy-Schwarz inequality:


    sum ai bi is less then or equal to sum (ai^2)^1/2 sum (bi^2)^1/2
     
  7. Mar 9, 2012 #6

    jbunniii

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    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    OK, good. Now, given that all the [itex]a_i[/itex] are nonnegative, what can you say about

    [tex]\sum a_i^2[/tex]

    if you know that

    [tex]\sum a_i[/tex]

    is finite?
     
  8. Mar 9, 2012 #7
    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    if ai converges (finite) then (ai)2 converges also.
     
  9. Mar 9, 2012 #8

    jbunniii

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    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    Correct, but do you know how to prove it?

    And can you see how to use this fact along with the Cauchy-Schwarz inequality to solve the problem?
     
  10. Mar 9, 2012 #9
    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    Since I am not sure how to use the Cauchy ineq. , how about something like this :

    Let
    Ai=Sum[i=0 to inf] ai
    Bi=Sum[i= 0 to inf] bi
    Ci=Sum[i=0 to inf] ai* bi
    Ai, Bi, Ci are obviously strictly increasing (1) , because
    Ai=A_(i-1)+a_n, similary Bi and Ci.
    Let lim(n->inf)Ai=X, lim(n->inf)Bi=Y (because they converge).
    Because they are strictly increasing,
    =>Ai=X and Bi=Y for every i.
    Ci=Ai*Bi, because
    a1*b1+a2*b2+...an*bn<(a1+a2+..+an)*
    *(b1+b2+..+bn)
    From this, Ci=Ai*Bi=X*Y (2)
    From (1) and (2) (monotonous and limited) Ci is convergent
     
  11. Mar 9, 2012 #10

    jbunniii

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    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    These definitions don't make any sense. On the right hand side, you have summed over i = 0 to infinity. The result therefore does not depend on i.

    Since your definition of Ai, Bi, Ci above doesn't actually depend on i, this statement also makes no sense. A constant can't be strictly increasing.

    Let's look at the Cauchy-Schwarz inequality again, which looks like the following assuming that [itex]a_i[/itex] and [itex]b_i[/itex] are non-negative:

    [tex]\sum a_i b_i \leq \sqrt{\sum a_i^2} \sqrt{\sum b_i^2}[/tex]

    Therefore, if [itex]\sum a_i^2[/itex] and [itex]\sum b_i^2[/itex] are finite, then so is [itex]\sum a_i b_i[/itex].

    We know that [itex]\sum a_i[/itex] and [itex]\sum b_i[/itex] are finite. If you can show that this implies that [itex]\sum a_i^2[/itex] and [itex]\sum b_i^2[/itex] are finite, then you're done. So focus on this step.

    Here's a hint: if [itex]x[/itex] is a nonnegative real number, what has to be true of [itex]x[/itex] in order to have [itex]x^2 \leq x[/itex]?
     
  12. Mar 9, 2012 #11

    Dick

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    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    Actually you don't really need Cauchy-Schwarz do you? If the series ai converges then ai approaches 0. So ai<1 for i large enough, doesn't it? Use a comparison test.
     
  13. Mar 10, 2012 #12

    jbunniii

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    Re: prove or give a counter example is sum ai and sum bi are convergent series with n

    Yes, that's much better. When I see an inner product I always think "Cauchy-Schwarz" but in this case it meant that I didn't notice the more direct proof.
     
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