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!!

LambLamb
Dec 1, 2015

#5**+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

#2**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!

LambLamb
Dec 1, 2015

#4**0 **

I have drawn attention to your question here

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

Melody
Dec 1, 2015

#5**+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