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Infinite Series Intuition

  1. May 13, 2014 #1
    Hello!

    How can I justify that the infinite series 1 - 1 + 1 - 1 + 1 - 1.... is divergent?

    If I were to look at this, I see every two terms canceling out and thus, and assume that it is convergent since the sum doesn't blow up. That's what my intuition would tell me.

    I know I can use different tests to figure out that it is divergent, but I don't have an intuition for why it's so.

    Any ideas? Thanks!
     
  2. jcsd
  3. May 13, 2014 #2

    micromass

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    By definition, a series ##\sum a_n## is convergent if the sequence

    [tex]a_1,~a_1+a_2,~a_1+a_2+a_3,~a_1+a_2+a_3+a_4,...[/tex]

    is convergent.

    So in your case, you have to investigate the sequence

    [tex]1,~1-1,~1-1+1,~1-1+1-1,~1-1+1-1+1,...[/tex]

    Is this sequence convergent?
     
  4. May 13, 2014 #3
    So you're saying that for a series to converge, it's the series of partial sums (that's the correct term right?) must also converge?

    And it's just alternating between the values 0 and 1 infinitely. So yeah it is divergent.

    Thanks so much.
     
  5. May 13, 2014 #4

    micromass

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    Yes, the sequence of partial sums must converge. That's the definition of when a series converges.
     
  6. May 13, 2014 #5

    Mark44

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    For a given series ##\sum_{n = 0}^{\infty}a_n##, there are two sequences that are involved:
    The sequence of terms in the series: {a0, a1, a2, ... , an, ...}.
    The sequence of partial sums: {a0, a0 + a1, a0 + a1 + a2, ... }.

    As micromass already said, if the sequence of partial sums converges to a number, then the series itself converves to that same number.

    Note that I showed a series that starts with an index of 0. The starting index can be some other integer.
     
    Last edited: May 13, 2014
  7. May 13, 2014 #6
    awesome, thanks so much guys!
     
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