Harmonic Series - How does this diverge?

[brushman]

I have read the proofs and understand them. But I don't get how it is possible that this diverges.

We know

\lim_{n\rightarrow\infty} 1/n = 0

So, given the harmonic series, aren't you eventually adding 0? I know we are not, but why not?

Apparently, the sequence is not approaching 0 'fast enough', but what determines how fast you need to approach 0 for a series to converge? And why is this an issue when we're looking at the infinite term?

 

[hgfalling]

Well, no, you aren't eventually adding zero, you're adding elements arbitrarily close to zero. But you're adding an arbitrarily large number of them.
Now sometimes, for example, if your terms are \frac{1}{n^2}, they get smaller fast enough that the sequence of partial sums converges. Other times (as in the harmonic series), they don't get small enough fast enough, so the sum diverges.

The question of "how fast do they have to go to zero?" is basically the question behind all those tests you're probably either studying right now or will study in the future. (Ratio, root, etc.)

 

[HallsofIvy]

In order that a series converge, the individual terms must go to 0. But that is a "necessary" condition, not "sufficient" condition. In fact, the individual terms must go to 0 fast enough.

One reason we pay attention to the "harmonic series" is that it is right on the boundary- if the terms of series go to 0 faster than 1/n, then the series converges. If not, then the series diverges.

 

[╔(σ_σ)╝]

The sequence of partial sums is not cauchy.

 

[brushman]

Why does the rate it goes to 0 even matter since we're looking at infinity? At infinity the limit is 0, so at infinity we should be adding 0 every time, thus it seems like the series would converge...