proof that Euler’s constant exists∗
rm50†
2013-03-21 21:49:39
Theorem 1 The limit
γ = lim
n→∞
n
X
1
− ln n
k
!
k=1
exists.
Proof. Let
Cn =
1
1 1
+ + · · · + − ln n
1 2
n
and
Dn = Cn −
Then
1
n
1
1
Cn+1 − Cn =
− ln 1 +
n+1
n
and
Dn+1 − Dn =
1
1
− ln 1 +
n
n
Now, by considering the Taylor series for ln(1 + x), we see that
1
1
1
< ln 1 +
<
n+1
n
n
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 bound for Cn . Thus the Cn are monotonically decreasing and
bounded below, so they must converge.
∗ hProofThatEulersConstantExistsi
created: h2013-03-21i by: hrm50i version: h38771i
Privacy setting: h1i hProofi h40A25i
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1
References
[1] E. Artin, The Gamma Function, Holt, Rinehart, Winston 1964.
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