Closed form of harmonic series

In summary, the conversation is about whether there is a proof that the harmonic sum \sum\frac{1}{k} has no closed form expression, similar to the proof for equations with degrees higher than 4. The series diverges to infinity and there is currently no known closed form expression for it. It is unclear if a closed form exists or not.
  • #1
Dansuer
81
1
HI!

I was wandering if there is a proof that the harmonic sum [itex]\sum\frac{1}{k}[/itex] has no closed form. Something like the proof that an equation with degree more than 4 has no solution in terms of radicals.
 
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  • #2
Dansuer said:
HI!

I was wandering if there is a proof that the harmonic sum [itex]\sum\frac{1}{k}[/itex] has no closed form. Something like the proof that an equation with degree more than 4 has no solution in terms of radicals.
This series diverges to infinity.
 
  • #3
yeah that's cool but that's not what I'm looking for. Maybe I've not been very clear.
I'll try again.

There is not a closed form expression of the harmonic sum [itex]\sum^{n}_{0}\frac{1}{k}[/itex]. which means it cannot be expressed in terms of elementary functions (e^x,sin(n), log(n), ...).
This is a closed form for [itex]\sum^{n}_{0} k [/itex]

[itex]\sum^{n}_{0} k = \frac{n(n+1)}{2}[/itex]

Does a closed form exist, but it's not yet been found ?
or it had been proved that it cannot exist ?
or maybe there is, maybe not, nobody knows anything about it ?
 

1. What is the closed form of the harmonic series?

The closed form of the harmonic series is ln(n), where n is the number of terms in the series. This means that as n approaches infinity, the sum of the series approaches the natural logarithm of n.

2. How is the closed form of the harmonic series derived?

The closed form of the harmonic series can be derived using the Euler-Maclaurin summation formula, which is a method for approximating the sum of a series. By applying this formula to the harmonic series and simplifying, we can arrive at the closed form of ln(n).

3. What is the significance of the closed form of the harmonic series?

The closed form of the harmonic series has many applications in mathematics and physics. It is used in the analysis of algorithms, in calculating the expected number of comparisons in sorting algorithms, and in the study of infinite series and their convergence.

4. Can the closed form of the harmonic series be proven?

Yes, the closed form of the harmonic series can be proven using mathematical induction. By showing that the formula holds true for n = 1 and then assuming it holds true for n = k, we can use algebraic manipulations to show that it also holds true for n = k + 1.

5. Are there any limitations to the closed form of the harmonic series?

While the closed form of the harmonic series is a useful tool in mathematics, it has its limitations. It only applies to the standard harmonic series where the terms are 1, 1/2, 1/3, 1/4, .... It cannot be used for other variations of the harmonic series, such as the alternating harmonic series. Additionally, the closed form only gives an approximation and the actual sum of the series can differ by a small amount.

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