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I have here the known series of natural logarthem.

$$\ln \left( \frac {1}{1-x} \right)=x+\frac {x^2}{2}+\frac {x^3}{3}+\frac {x^4}{4}+\frac {x^5}{5}+\cdots \tag 1$$

and

$$\ln(1+x)=x-\frac {x^2}{2}+\frac {x^3}{3}-\frac {x^4}{4}+\frac {x^5}{5}-\cdots \tag 2$$

Subtract $(2)$ from $(1)$

$$\ln\left(\frac {1}{1-x}\right)-\ln(1+x)=x^2 +\frac {x^4}{2}+\frac {x^6}{3}+\frac {x^8}{4}+\cdots \tag 3$$

Now if we let $x \to 1$

$$\lim_{x \to 1}\left[\ln\left(\frac 1{1-x}\right)-(x^2 +\frac {x^4}{2}+\frac {x^6}{3}+\frac {x^8}{4}+\cdots)\right]=\ln (2) \tag 4$$

So what is the difference between this and Euler–Mascheroni constant limit?

$$\gamma=\lim_{n \to\infty} \left(-\ln (n)+\sum_{k=1}^{n} \frac 1 k \right) \tag 5$$

I am sure that $\gamma \neq -\ln(2)$

There must be a problem that I can not recognize.

Michael Hardy
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Pentapolis
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1 Answers1

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The problem is that you have four different quantities all diverging as $n\to\infty$ and $x\to1^-$. Whether the difference between any two of these diverges or converges to any particular value depends on exactly how each one diverges.

You can make $\ln \left( \frac {1}{1-x} \right)$ and $ln(n)$ diverge in exactly the same way by setting them equal, that is, by setting $x = 1 - \frac1n$. Now compare $\sum_{k=1}^{n} \frac 1 k$ with $x^2 +\frac {x^4}{2}+\frac {x^6}{3}+\frac {x^8}{4}+\cdots$ for various increasing values of $n$. You may notice that when $n = 1$, $\sum_{k=1}^{n} \frac 1 k - \left(x^2 +\frac {x^4}{2}+\frac {x^6}{3}+\frac {x^8}{4}+\cdots\right) = 1$, and that for larger values of $n$ this difference does not approach zero but rather continues to grow.

This "excess" value of $\sum_{k=1}^{n} \frac 1 k$ explains why $\gamma > -\ln(2)$.

David K
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  • It seems to me For larger values of $n$ , $x \to 1$ which reduce the difference between the two series, and also the value of $x < 1$ always then the growth is going down. the picture still not clear to me. thank you for your answer +1. – Pentapolis Sep 03 '16 at 01:57
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    Both series grow infinitely large through very different mechanisms, so any idea that the difference between them should ever approach zero is a mistaken intuition. On the contrary, your own calculations indicate that the difference should approach a number slightly larger than $1.27$. – David K Sep 05 '16 at 03:06