Harmonic series, slowest diverging series?

In summary, the harmonic series is divergent and very slow to diverge. It is not the slowest diverging series, as there are ways to construct even slower diverging series. The idea of a least divergent series is not possible, as there will always be ways to slow a series even more. For example, the prime harmonic series has a slower divergence rate than the harmonic series.
  • #1
soothsayer
423
5
The harmonic series is divergent, and in general, I know that just because one series is larger than another divergent series, doesn't mean the series is convergent. However, the harmonic series is very, very slow to diverge. Is the harmonic series the slowest diverging series? That is, is it the case that any series larger than the harmonic series must necessarily converge and is there any easy proof to show this?
 
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  • #2
1+1/2+1/4+1/6+... is a counter example. Proof: 1/4+1/6+1/8 > 1/2

In general, the larger n is, in 1+1/2+1/(2+n)+1/(2+2n)+..., the slower the divergence.
EDIT: Even more general, the faster the arithmetic series in the denominator diverges, the slower the series diverges.

This is an interesting question. Congrats.
 
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  • #3
The harmonic series is "exponentially slow" to diverge, in the sense that
2 terms (1/3 + 1/4) sum to > 1/2
the next 4 terms sum to > 1/2
the next 8 terms sum to > 1/2
and so on for blocks of 2^k terms.

You could construct a slower divergiing series, for example one where
2 terms sum to > 1/3
4 terms sum to > 1/4
8 terms sum to > 1/5
16 terms sum to > 1/6
and so on.

And then construct an infinite set of series, each diverging slower than the previous one, by repeating this process.

Or to put it another way, the first N terms of the harmonic series sums to approximately log(N), but you could construct slower diverging series where the first N terms sum to approximately log(log(N)), log(log(log(N))), and so on.
 
  • #4
Thanks, yeah, that makes sense.

The reason I was actually asking was because I was looking at the infinite series of 1/3ln(n) and recognized it to be smaller than the harmonic series, which would 1/eln(n). I was supposed to determine whether that first series converged or diverged and I couldn't figure out how to determine it, but I wondered, since it was smaller than the harmonic series, if it was necessarily convergent. I know now that it doesn't, but if you had an infinite series of the form 1/kln(n), for what k values is the sum convergent and why?
 
  • #5
The idea of a least diverget series is doomed. There will always be ways to slow such a series much more. Like AlephZero mentioned there is the famous prime harmonic series
1/2+1/3+1/5+1/7+1/11+1/13+1/17+1/19+...+1/n~log log n
where each denominator is prime
compare to the harmonic series
1/1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+...+1/n~log n
 

1. What is a harmonic series?

A harmonic series is a type of mathematical series that is formed by taking the reciprocal of each positive integer and adding them together. It is often represented as 1 + 1/2 + 1/3 + 1/4 + ...

2. How does a harmonic series diverge?

A harmonic series diverges because as the series goes on, the terms get closer and closer to 0, but never actually reach 0. This means that the sum of the series keeps getting larger and larger as more terms are added.

3. Why is the harmonic series considered the slowest diverging series?

The harmonic series is considered the slowest diverging series because it diverges at a slower rate than any other divergent series. This means that it takes the longest amount of time for the series to reach infinity.

4. How is the divergence of a harmonic series proven?

The divergence of a harmonic series can be proven using the integral test or the comparison test. The integral test involves taking the limit of the sum of the series as n approaches infinity and comparing it to the integral of 1/x. The comparison test involves comparing the series to another divergent series, such as the geometric series.

5. What are some real-world applications of harmonic series?

Harmonic series have various applications in physics, engineering, and economics. For example, in physics, harmonic series can be used to describe the oscillations of a vibrating string or the harmonics of a sound wave. In engineering, they can be used to analyze circuits or to model the vibration of structures. In economics, harmonic series can be applied to study the distribution of wealth or income.

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