PlanetMath: proof that Euler's constant exists

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proof that Euler's constant exists

(Proof)

Theorem 1 The limit

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

exists.

$\displaystyle C_n=\frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n$

and

$\displaystyle D_n=C_n-\frac{1}{n}$

Then

$\displaystyle C_{n+1}-C_n=\frac{1}{n+1}-\ln\left(1+\frac{1}{n}\right)$

and

$\displaystyle D_{n+1}-D_n=\frac{1}{n}-\ln\left(1+\frac{1}{n}\right)$

Now, by considering the Taylor series for $\ln(1+x)$ , we see that

$\displaystyle \frac{1}{n+1}<\ln\left(1+\frac{1}{n}\right)<\frac{1}{n}$

and so

$\displaystyle C_{n+1}-C_n < 0 < D_{n+1}-D_n$

Thus, the

decrease monotonically, while the

increase monotonically, since the differences are negative (positive for

). Further,

and thus

. Thus the

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"proof that Euler's constant exists" is owned by rm50.

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Cross-references: converge, bounded, monotonically decreasing, lower bound, positive, negative, differences, monotonically, Taylor series, proof, limit
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This is version 3 of proof that Euler's constant exists, born on 2007-01-15, modified 2007-04-15.
Object id is 8771, canonical name is ProofThatEulersConstantExists.
Accessed 1442 times total.

Classification:

40A25 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Approximation to limiting values )

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