proof that Euler’s constant exists


Theorem 1

The limit

γ=limn(k=1n1k-lnn)

exists.

Proof. Let

Cn=11+12++1n-lnn

and

Dn=Cn-1n

Then

Cn+1-Cn=1n+1-ln(1+1n)

and

Dn+1-Dn=1n-ln(1+1n)

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

1n+1<ln(1+1n)<1n

and so

Cn+1-Cn<0<Dn+1-Dn

Thus, the Cn decrease monotonically, while the Dn increase monotonically, since the differences are negative (positive for Dn). Further, Dn<Cn and thus D1=0 is a lower boundMathworldPlanetmath for Cn. Thus the Cn are monotonically decreasing and bounded below, so they must converge.

References

Title proof that Euler’s constant exists
Canonical name ProofThatEulersConstantExists
Date of creation 2013-03-22 16:34:48
Last modified on 2013-03-22 16:34:48
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Proof
Classification msc 40A25