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# Σ Infinite Sum

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+495

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

#5
+30019
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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
+8574
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Are you putting numbers (1,2,3,4,5) Into that equation for x?

Dec 1, 2015
#2
+495
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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
+109512
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I have drawn attention to your question here

Dec 1, 2015
#5
+30019
+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!!

Alan Dec 1, 2015
#6
+109512
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Thank you Alan.

Dec 1, 2015
#7
+495
0

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

Dec 2, 2015