Harmonic series

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Homework Statement


i know that k = 0 to∞∑(1/ k) is harmonic series( we know that the sum is divergent) , how about ∑(1/ k+1 ) ???

Homework Equations




The Attempt at a Solution


in my opinion , it's also harmonic series , because the sum is divergent . Am i right ?
 

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  • #2
Ray Vickson
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Homework Statement


i know that k = 0 to∞∑(1/ k) is harmonic series( we know that the sum is divergent) , how about ∑(1/ k+1 ) ???

Homework Equations




The Attempt at a Solution


in my opinion , it's also harmonic series , because the sum is divergent . Am i right ?
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.)
 
  • #3
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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.)
i mean second one , IMO , it is also harmonic .... , am i right ?
 
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Ray Vickson
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i mean second one , IMO , it is also harmonic .... , am i right ?
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.
 
  • #5
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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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