What's the relation (IMPORTANT)?

[TheDestroyer]

What's the relation (IMPORTANT)??

Hi guys, let me introduce my self, I'm a physicist studying in the university of aleppo...

I want the relation between those number as a continuous function, my head is going to blow ! i can't find it, can anyone help?

x->y

1->1

2->3/2

3->11/6

4->25/12

5->137/60

6->147/60

7->363/140

8->761/280

9->7192/2520

hint: I can see that every number (y) is multiplied by 1/x

PLEASE, I don't want any approximation, interpolation or anything like that because i can do it alone, i need an exact function

for example, beta, gamma, airy, fresnels s and c, hermite, legender, or anything else BUT PLEEEEASE AN EXACT FUNCTION,

I've attached the graph of this damn function

It's not an easy question, i don't think I'm receiving an answer before 10 years, lol !

Thanks Guys

 

[StatusX]

Those look like the harmonic numbers:

$$H_n = \sum_{k=1}^n \frac{1}{k}$$

I don't believe there is an explicit formula for the nth term, but I could be wrong.

EDIT: I'm sorry. Apparently there is one, although I don't know how useful it would be to you. See the mathworld article on harmonic numbers.

 

[TheDestroyer]

Thank you, but i wanted it as a CONTINUOUS FUNCTION, not as a sum, integrals can be accepted as i mentioned above, thanks anyway for the link, any other ideas?

 

[StatusX]

Well, as the link mentions, the Harmonic numbers are given by:

$$H_n = \gamma + \psi_0(n+1)$$

where $\gamma$ is the Euler-Masceroni constant and $\psi_0(x)$ is the digamma function, related to the gamma function $\Gamma(x)$ by:

$$\psi_0(x) = \frac{\Gamma'(x)}{\Gamma(x)}$$

So the function:

$$f(x)=\gamma + \psi_0(x+1)$$

Is a continuous function which has the Harmonic numbers as its value at the positive integers.

 

[D H]

Thank you, but i wanted it as a CONTINUOUS FUNCTION, not as a sum, integrals can be accepted as i mentioned above, thanks anyway for the link, any other ideas?

Did you look at the link? The answer you want is there.

 

[TheDestroyer]

Sorry, The time was 2 am, and i saved the link and went to bed, thank you guys, i got what i want

Thanks again

 

[Hurkyl]

I've edited the original post (and the subthread related to what I cut out) in the hopes that I don't have to lock this.

 

[TheDestroyer]

LOL ! Why should you Hurkyl?? we didn't do anything wrong, did we? it's just a discussion about sciences !