Harmonic Series - How does this diverge?

In summary, the issue with the harmonic series is that the terms do not approach 0 fast enough for the series to converge. This is determined by the rate at which the terms approach 0, which is a necessary condition for convergence. The harmonic series is right on the boundary, as if the terms approach 0 faster than 1/n, the series converges. However, since the terms do not approach 0 fast enough, the series diverges. The sequence of partial sums is also not cauchy. At infinity, although the limit is 0, the terms are not actually 0 and are instead arbitrarily close to 0, leading to the divergence of the series. Various tests, such as the ratio and root tests
  • #1
brushman
113
1
I have read the proofs and understand them. But I don't get how it is possible that this diverges.

We know

[tex] \lim_{n\rightarrow\infty} 1/n = 0[/tex]

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?
 
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  • #2
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 [itex] \frac{1}{n^2} [/itex], 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.)
 
  • #3
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.
 
  • #4
The sequence of partial sums is not cauchy.
 
  • #5
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...
 

Related to Harmonic Series - How does this diverge?

1. What is the harmonic series?

The harmonic series is an infinite series in mathematics that is defined as the sum of the reciprocals of the natural numbers (1, 2, 3, etc.). It is represented by the notation ∑n=1 1/n and is considered to be a divergent series.

2. How does the harmonic series diverge?

The harmonic series diverges because as the terms of the series increase, they approach infinity. This means that no matter how many terms are added, the series will never reach a finite value and will continue to grow indefinitely.

3. What is the significance of the harmonic series?

The harmonic series is important in mathematics as it is an example of a divergent series, which helps to understand the concept of convergence and divergence in infinite series. It also has applications in physics, particularly in the study of sound waves and musical intervals.

4. Can the harmonic series be manipulated to converge?

No, the harmonic series cannot be manipulated to converge. No matter how the terms are rearranged, the series will always diverge. This is known as the divergence of the harmonic series and was first proven by the mathematician Nicole Oresme in the 14th century.

5. How is the divergence of the harmonic series related to the concept of infinity?

The divergence of the harmonic series is related to the concept of infinity because as the terms of the series approach infinity, the sum of the series also approaches infinity. This illustrates the idea that infinity is not a finite number and cannot be reached or calculated in the same way as other numbers.

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