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Alternating series question

  1. Mar 18, 2015 #1

    joshmccraney

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    Hi PF!

    The other day I was showing convergence for an alternating series, let's call it ##\sum (-1)^n b_n##. I showed that ##\lim_{n \to \infty} b_n = 0## and that ##b_n## was monotonically decreasing; hence the series converges by the alternating series test. but I needed also to show it did not converge to zero. the argument I used was that since ##|b_1 - b_2| >0## and that since ##b_n## monotonically decreases, we then know ##\sum (-1)^n b_n > |b_1 - b_2|##. Is my intuition correct here? If so, is it ever possible to have a series described above converge to zero?

    Let me know what you think!
     
  2. jcsd
  3. Mar 18, 2015 #2

    wabbit

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    First thought this was wrong, but yes.

    Set ## a_n=b_{2n}-b_{2n+1}\geq 0 ##

    ## \sum(-1)^n b_n =\sum a_n \geq 0 ## and this can only be zero if ## \forall n, b_n=0 ##

    Edit you seem to have the index n starting at 1 instead of 0 so you need to adjust the above a little, but this doesn't change anything except perhaps a sign.
    And your argument, assuming the b's are strictly decreasing, works - you can also complete it to cover all cases where b is not identically 0.
     
    Last edited: Mar 19, 2015
  4. Mar 19, 2015 #3

    joshmccraney

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    Thanks! I thought so but wanted reassurance.
     
  5. Mar 19, 2015 #4

    mathman

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    Wrong conclusion! [itex]a_n=0[/itex] will hold if pairs of alternate terms have the same magnitude [itex]b_{2n}=-b_{2n+1}[/itex], and [itex]b_{2n} \gt b_{2n+2}[/itex].
     
  6. Mar 19, 2015 #5

    joshmccraney

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    But ##b_n## is monotonically decreasing.
     
  7. Mar 19, 2015 #6

    wabbit

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    Oops you are right of course !

    The sum is 0 iff ## \forall n, b_{2n}=b_{2n+1} ## and that's all we can say.

    Of course if we know that b is strictly dereasing this cannot happen, but i was trying to avoid needing that since op did not say "strict".

    Failed attempt, sorry.

    To expand, the generic counterexampe to my initial claim is the altermating sum
    ## b_0-b_0+...+b_n-b_n+...##
    which converges to 0 iff ## b_n\rightarrow 0 ##
     
    Last edited: Mar 19, 2015
  8. Mar 22, 2015 #7
    I'd use comparison test with another series that has similar rate of change as the one you're using.

    ##\sum \limits_{i} \left | {b_i} \right | \geq \sum \limits_{i} \left | {a_i} \right |##

    Where ##a_i \approx b_i## in structure, but ##a_i## is both monotonic and bounded.
     
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