An Approach to Mathematic Functions Basics

by Frank C. Fung ( 1st published in December, 2014. )

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Section VII - The Logarithms of Natural Numbers

Summary for the Section :

• In this Section , we shall attempt to express :
  • the Natural Logarithm of any { Positive Integer greater than [ 2 ] }
    • via the multiple use of Infinite Series .

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The core Infinite Serires for Natural Logarithms :

• Let us bring-in again the results from the last Section for the function [ PHI-of-(x) ] :
  • 

    where :

    • the [ PHI-of-(x) ] is any function that qualifies as a Logarithmic Function .

• For Natural Logarithms , the value of [ K ] is set equal to [ 1 ] .

  Consequently , the Expansion Formula for Natural Logarithms based on the above is then :

  • 

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The Natural Logarithm of [ 2 ] :

• We note here that :
  • for the natural number [ 2 ] :
    • [ 2 ] = 1 + 1

    and

    • we can plug this into the above Expansion Formula to arrive at the { Natural Logarithm of [ 2 ] } .

  We then have :

  • 

  Therefore :

  • 

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The Expansion Formula using Recipricals :

• Let us now take a quick look at the case where [ x ] is a reciprical .

  And we set up :

  • [ x ] = 1 / n

    where [ n ] is any [ positive integer ] .

  Let us now bring back the original Expansion Formula for Nature Logarithms from above ,

  • i.e. :
    • 

  and

  • plug-in the value [ x = 1 / n ] to arrive at this equation :
    • 

• And we shall put this formula to good use immediately below .

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The Natural Logarithms of { Positive Integers greater than [ 2 ] } :

• Let [ X ] be any positive integer greater than [ 2 ] .

  We can then write :

  • 

  Further munipulations then yield us this relation :

  • 

• Taking logarithms on both sides then yields us this equation :
  • 

  And we take note at this point that :

  • 

  Consequently :

  • 

• Let us now bring-in from above the { Expansion Formula using Recipricals } , i.e. :
  • 

  Applying this to the equation immediately above then yields us this relation :

  • 

• Let us now bring in a dummy variable [ delta ] such that :
  • 

  And consequently :

  • 

  Thus , we can now re-write the equation above in this special format :

  • 

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Bringing-in the Zeta Sums :

• Let us bring-in , at this point , the Zeta Sums , i.e. :
  • 

  And the author's prefered format for the same is :

  • 

  We take note here that :

  • based on the work of Euler ,

    we do have these rather famous results below for the Zeta Sums :

    • 

      and

    • 

    • 

    • 

  And we can also confirm that the Zeta Sums do converge for s = 3, 5 , 7 , etc.

• Let us bring back our formula above for the { logarithms of Natural Numbers } :
  • 

  And we see here that :

  • the Zeta Sums on the right-hand-side do converge from the 2nd term onwards .

  And this is something that we should be keeping in mind ,

  • before moving forward on to the Riemann Zeta Function :
    • 

      involving [ complex values ] for [ s ] .

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go to the next section : Section VIII - A Statement of Purpose for Part III

go to the last section : Section VI - Expressing Logarithms via an Infinite Series

return to the MATH Functions Basics HomePage

Original dated 2014-12-07