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Harmonic series

  1. Apr 13, 2016 #1
    1. The problem statement, all variables and given/known data
    i know that k = 0 to∞∑(1/ k) is harmonic series( we know that the sum is divergent) , how about ∑(1/ k+1 ) ???

    2. Relevant equations


    3. The attempt at a solution
    in my opinion , it's also harmonic series , because the sum is divergent . Am i right ?
     
  2. jcsd
  3. Apr 13, 2016 #2

    Ray Vickson

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    Note: the series cannot be ##\sum_{k=0}^{\infty} 1/k## because the first term would be 1/0. However, starting at ##k = 1## is OK.

    Do you mean that one of the series is ##\sum_{k=1}^{\infty} 1/k## and the other is ##\sum_{k=1}^{\infty}(\frac{1}{k}+1)##? That is what you wrote. Did you really mean ##\sum_{k=1}^{\infty} 1/(k+1)## for the second series? If so, use parentheses, like this: 1/(k+1).

    Anyway, just write out the first first few terms of both of your series, to see how they are related.

    When you say "it's also harmonic series , because the sum is divergent" you have it backwards: it is not harmonic because it is divergent; it is divergent because it is harmonic. (Lots of divergent series are not at all harmonic.)
     
  4. Apr 13, 2016 #3
    i mean second one , IMO , it is also harmonic .... , am i right ?
     
  5. Apr 13, 2016 #4

    Ray Vickson

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    Have you tried to write out the first few terms of both series to see how they differ? If you do, you can answer your own question.
     
  6. Apr 13, 2016 #5

    Math_QED

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    The infinite sum of 1/(k+1), with k starting from 1, is the same as the infinite sum of 1/k, with k starting from one minus 1. Follow Ray Vickson's advice to see this.
     
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