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Does anybody know of a function that describes the infinite sequence 1/1 + 1/2 +1/3 +1/4 + 1/5 ... etc?

 

For 1 + 2 + 3 + 4 + 5 ... etc, I know about x(x+1)/2, but is there an analogous function for the infinite sequence above?

 

I tried working it out, but I can't figure out what goes on the numerator. And I know it slowly tends toward infinity. Any help is appreciated!!

 Dec 1, 2015

Best Answer 

 #5
avatar+28182 
+10

This is known as the harmonic series and the partial sum is given by:

 

\(\Sigma_{k=1}^n\frac{1}{k}=\gamma+\psi_0(n+1)\)

 

where \(\gamma\)  is the Euler-Mascheroni constant (0.5772....)

 

and \(\psi_0(x)\)  is the digamma function

 

(see http://mathworld.wolfram.com/HarmonicSeries.html)

 

This is far from elementary mathematics!!

 Dec 1, 2015
 #1
avatar+8613 
0

Are you putting numbers (1,2,3,4,5) Into that equation for x?

 Dec 1, 2015
 #2
avatar+495 
0

Yes. here's a T-table of x's and y's

 

x y (decimal expansion)
1 1 /1 (1.000000...)
2 3 /2 (1.500000...)
3 11 /6 (1.833333...)
4 50 /24 (2.083333...)
5 274 /120 (2.283333...)
6 1764 /720 (2.45...)
7 13068 /5040 (2.592857...)
8 109584 /40320 (2.717857...)
9 1026576 /362880 (2.828968...)

 

So I figured out the x! goes on the denominator. I'm not sure about the numerator.

 

But I noticed this pattern: to get the numerator for y, take the numerator of (y-1), multiply it by x, then add (x-1)!

 

Any help is appreciated!

 Dec 1, 2015
edited by LambLamb  Dec 1, 2015
edited by LambLamb  Dec 1, 2015
edited by LambLamb  Dec 1, 2015
 #4
avatar+105606 
0

I have drawn attention to your question here

 

http://web2.0calc.com/questions/unanswered-questions-that-stand-out-for-some-reason

 Dec 1, 2015
 #5
avatar+28182 
+10
Best Answer

This is known as the harmonic series and the partial sum is given by:

 

\(\Sigma_{k=1}^n\frac{1}{k}=\gamma+\psi_0(n+1)\)

 

where \(\gamma\)  is the Euler-Mascheroni constant (0.5772....)

 

and \(\psi_0(x)\)  is the digamma function

 

(see http://mathworld.wolfram.com/HarmonicSeries.html)

 

This is far from elementary mathematics!!

Alan Dec 1, 2015
 #6
avatar+105606 
0

Thank you Alan.   laugh

 Dec 1, 2015
 #7
avatar+495 
0

Thank you so much! I don't know if I could have done all that on my own.

 Dec 2, 2015

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