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## Homework Statement

Prove that the [tex]\sum1/n[/tex] is divergent.

Does anyone know a simple proof for this. I understand that it does not converge intuitively but I'm not sure how to prove it in symbols.

Thank you for your help.

M

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- Thread starter michonamona
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- #1

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Prove that the [tex]\sum1/n[/tex] is divergent.

Does anyone know a simple proof for this. I understand that it does not converge intuitively but I'm not sure how to prove it in symbols.

Thank you for your help.

M

- #2

rock.freak667

Homework Helper

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The integral test works well for that one.

- #3

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- #4

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- #5

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- #6

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H = \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...

[/tex]

Notice that 1/3 + 1/4 > = 1/4 + 1/4 = 1/2.

And that that 1/5 + 1/6 + 1/7 + 1/8 > 1/8 + 1/8 + 1/8 + 1/8 = 4/8 = 1/2

So, in essence,

[tex]

H > 1 + \sum_{n=1}^{\infty} \frac{1}{2}

[/tex]

The latter part diverges, of course.

EDIT: This is the intuitive way. There is also an elementary, but rigorous way of doing it without the integral test.

- #7

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2^k*a_2^k

<a_(2^k)+...

+a_(2^(k+1)-1) , by summing over these you Get series comparison

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