Showing harmonic series is divergent

  • Thread starter brycenrg
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  • #1
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Homework Statement


image.jpg


Homework Equations


Where do the terms 1/4 come from? Are they ambiguous?

The Attempt at a Solution


Trying to understand the text[/B]
 

Answers and Replies

  • #2
RUber
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The 1/4 is a convenient comparison since 1/3 > 1/4.
By comparing the series against something easy to calculate, you can show that the series is growing and not converging.
 
  • #3
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I'm not to familiar with proofs. But I can say any number? Why don't I just use 1/100? That's obviously smaller
 
  • #4
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By showing that the left side is greater than the right what does that show?
 
  • #5
RUber
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You are trying to show that the series does not converge.
In order to do that you are replacing some terms with smaller terms to make a known series that does not converge. Therefore the series in question must not converge because it is bigger than another series that doesn't converge.
The pattern you are supposed to see is that :
## \sum_{n=1}^{2^k} 1/n>{k}##
 
  • #6
RUber
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If you wanted to replace the terms with 1/100, you would need a lot more terms to see the pattern, but you could still notice that the first hundred terms would be greater than 1, the next 10000 terms would be greater than 1, so the sum would be more than two, the next million terms would be more than 1, for a total sum greater than 3 and so on. The point is that no matter how small the fraction, you won't run out of numbers needed to show you can add to another whole number.
 
  • #7
HallsofIvy
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The easiest way to show that the harmonic series does not converge is to use the "integral test". It is easy to show that [tex]\int_1^\infty \frac{1}{x} dx[/tex] does not exist so the harmonic series does not converge.
 

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