Why Is There No Simple Formula for the Sum of a Harmonic Progression?

[parshyaa]

What's the reason which implies that we can't have a formula for the sum of HP. https://en.m.wikipedia.org/wiki/Harmonic_progression_(mathematics)
Wikipedia gave a reson , can you elaborate it.

 

[jedishrfu]

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]

 

[parshyaa]

Do you mean the phrase:

Yes

 

[jedishrfu]

I found this discussion and proof online that you'll have to work through to understand:

http://math.stackexchange.com/quest...f-that-sum-limits-k-1n-frac1k-is-never-an-int

 

[parshyaa]

I think this video makes every thing clearer, thanks jedishrfu

 

[parshyaa]

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

 

[Igael]

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.

 

[Svein]

As n grows large, you have $\sum_{k=1}^{n}\frac{1}{k}\approx \ln(n)+\gamma$.

 

[Igael]

there is a nice but not very useful formula :
$\sum_{k=1}^{n}{\frac{1}{{a}+{b k}}}=\frac{{\psi^{(0)}({{\frac{a}{b}}+{n}}+{1})}-{\psi^{(0)}({\frac{a}{b}}+{1})}}{b}$ where $\psi^{(n)}(u)$ is the polygamma function

 

[parshyaa]

Are you asking why there is not a smart formula for the exact sum with k from 0 to n ?

Yes