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Tuesday, December 13, 2011

A New Formula for the Natural Logarithm of a Natural Number

I must share with you my pride and pleasure that I have the right to post here the following article of one of my lecturers, Dr. S. Nevo who research calculus and his doctorate was about the famous Zalcman's Lemma


Abstract. For every natural number T; we write Ln T as a series, generalizing the
known series for Ln 2.


  1. Richard Pennington asked me to post this comment because for technical reason he could not comment by himself.

    "Please pass on my comment to Dr. Nevo.

    Incidentally, I was taught at age 16 the following (for |x| < 1):

    ln (1+x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 ...
    ln (1-x) = -x - x^2/2 - x^3/3 - x^4/4 - x^5/5 - ...


    ln ((1+x)/(1-x)) = 2 (x + x^3/3 + x^5/5 + ...)

    Now set x = 2/3 and the left side is ln(5). And the series is absolutely convergent."

    Richard Pennington

  2. Dr. Shahar Nevo asked me to reply to Richard, the following:
    "Dear Richard,
    This is very impressive for the age of 16. I don't know how old are you now, but however,the formula you stated was discovered long ago you were born.The formula I suggest is not a power series but some arrangement of two harmonic
    series: one as is and one with minuses. It gives one more justification to call e "the basis of natural numbers".
    All the best
    Dr. Shahar Nevo"

  3. Hello Steve and Dr. Nevo. In answer to the previous comment, I am now 52.

    I had a play with the formulae in Dr. Nevo's paper and it all works fairly well (albeit with rather slow convergence). In particular, there are no problems with the convergence, which was my primary concern.

    I did have reservations about rearranging the terms of the harmonic series, as the series is only conditionally convergent, and there is a danger of incorrect results being derived. See, for example, pages 125-130 of Eugene P. Northrop's "Riddles in Mathematics" (English Universities Press 1944; Pelican Books, 1960-1974, ISBN 0 14 02 0478 4), in a chapter titled "Paradoxes of the Infinite". Incidentally, my copy indicates that I was awarded it as a school prize in 1974, at the age of 15.

    My experiments show that the error terms in your equation (3) and in your "alternative" sum for 0 both decrease proportionally to 1/N, where N is the number of terms in the sum. By contrast, the error terms in the expression I gave in my first posting (via Steve Solun, with thanks), decrease geometrically as the number of terms increases. The effect is clear: to calculate ln(5) with an error of 1/1000 takes 200 brackets (1000 terms) with your equation (3), but only up to the x^13/13 term with the expression in my first posting.

    It is also interesting to note that in the expression for zero, the first term is 1/12, and all the other terms are negative.

    For the reasons given above, I prefer the power series for calculations, but note that there are some interesting derivations which arise from your approach.

    Apologies for posting as Anonymous, but not having any of the accounts (Google Account, Wordpress, etc.) available, and not having my own website (the system doesn't like an email address as a URL), this is the only remaining option.

    All the best,
    (Dr.) Richard Pennington

  4. Will transfer to Dr.Nevo.
    It could be interesting if you both could write an article combining some interesting results, what do you think?



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