Sum of harmonic progression

In summary, the reason we cannot have a formula for the sum of a harmonic progression is that at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator, making it impossible to sum to an integer. This is discussed and proven in various online sources, such as the given Wikipedia page and a Math Stack Exchange discussion. While there are some formulas for specific cases, there is no general formula for the exact sum of a harmonic progression with k from 0 to n.
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  • #2
Do you mean the phrase:

It is not possible for a harmonic progression (other than the trivial case where a = 1 and k = 0) to sum to an integer. The reason is that, necessarily, at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator.[1]
 
  • #3
jedishrfu said:
Do you mean the phrase:
Yes
 
  • #5
I think this video makes every thing clearer, thanks jedishrfu
 
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  • #6
Still i don't have any clue/answer for why there is no formula for sum of HP for n terms, and i am not able to open your link
 
  • #7
Are you asking why there is not a smart formula for the exact sum with k from 0 to n ? The wiki page speaks mainly of integer sum and the clever video of divergence and infinite sum, which are another things.
 
  • #8
As n grows large, you have [itex]\sum_{k=1}^{n}\frac{1}{k}\approx \ln(n)+\gamma [/itex].
 
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  • #9
there is a nice but not very useful formula :
[itex]\sum_{k=1}^{n}{\frac{1}{{a}+{b k}}}=\frac{{\psi^{(0)}({{\frac{a}{b}}+{n}}+{1})}-{\psi^{(0)}({\frac{a}{b}}+{1})}}{b}[/itex] where [itex]\psi^{(n)}(u)[/itex] is the polygamma function
 
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  • #10
Igael said:
Are you asking why there is not a smart formula for the exact sum with k from 0 to n ?
Yes
 

What is a harmonic progression?

A harmonic progression is a sequence of numbers in which the reciprocal of each number is in arithmetic progression. In other words, the difference between each consecutive reciprocal is constant.

How do you find the sum of a harmonic progression?

The sum of a harmonic progression can be found by using the formula: S = (1/a) * (n/d) * (2n + (d-1)a), where S is the sum, a is the first term, n is the number of terms, and d is the common difference between the reciprocals.

What is the significance of harmonic progression in mathematics?

Harmonic progression has various applications in mathematics, including in number theory, calculus, and physics. It is also used to model real-life phenomena such as sound waves and electrical circuits.

Can the sum of a harmonic progression be infinite?

Yes, the sum of a harmonic progression can be infinite if the terms become smaller and smaller without approaching zero. This is known as a divergent harmonic progression. However, if the terms approach zero, the sum will converge to a finite value.

Are there any real-life examples of harmonic progression?

Yes, harmonic progression can be seen in various natural phenomena, such as the vibrations of a guitar string, the frequencies of musical notes, and the oscillations of a pendulum. It is also used in financial calculations, such as calculating compound interest over time.

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