Assume S is a finite sum.

Group it as:

S = 1 + [(1/2 + 1/3 + 1/4) + (1/5 + 1/6 + 1/7) + (1/8 + 1/9 + 1/10) + ...] > **

1 + [3(1/3) + 3(1/6) + 3(1/9) + ...] =

1 + [1 + 1/2 + 1/3 + ...] =

1 + S

But this has S > 1 + S.

This is impossible if S is to have a finite value.

So this is a contradiction, and therefore S must

not have a finite sum.

That is, S is divergent.

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I will start with:

Again, assume S is a finite sum.

S = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ...

S = 1 + (1/2 + 1/3) + (1/4 + 1/5) + (1/6 + 1/7) + (1/8 + 1/9) + ...

S = 1 + (5/6) + (9/20) + (13/42) + (17/72) + ...

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Prove that the Harmonic series is divergent.

Let S = the Harmonic series.

Assume S has a finite value.

S = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...

S = (1 + 1/2) + (1/3 + 1/4) + (1/5 + 1/6) + ...

> (2/2) + (2/4) + (2/6) + ...

= 1 + 1/2 + 1/3 + ...

= S

But this states that S > S, which is impossible,

because S is assumed to be finite.

So, this is a contradiction, and the Harmonic series

is divergent.

Wikipedia site (in the top first few pages), use an alternate

method to prove the divergence of the Harmonic series:

http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)

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