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Euler-Mascheroni Constant
Assuming "Euler-Mascheroni Constant" is referring to a mathematical definition | Use as
a mathematical constant
instead
Input interpretation
Euler-Mascheroni constant
Alternate names
Euler constant
|
Euler’s constant
|
Mascheroni constant
Basic definition
The Euler-Mascheroni constant is the mathematical constant defined as the limit of the difference between the nth partial sum of the harmonic series and the natural logarithm of n which has value of approximately 0.577.
Detailed definition
Fewer details
The Euler-Mascheroni constant
γ
, sometimes also called ‘Euler’s constant’ or ‘the Euler constant’ (but not to be confused with the constant
e2.718281…
) is defined as the limit of the sequence
γ
=
lim
n∞
n
∑
k1
1
k
-lnn
=
lim
n∞
(
H
n
-lnn),
where
H
n
is a harmonic number . It was first defined by Euler, who used the letter
C
and stated that it was “worthy of serious consideration” (Havil 2003, pp. xx and 51). The symbol
γ
was first used by Mascheroni.
γ
has the numerical value
γ0.577215664901532860606512090082402431042…
(OEIS A001620), and is implemented in the Wolfram Language as EulerGamma.
Related terms
Euler-Mascheroni constant approximations
|
Euler-Mascheroni constant continued fraction
|
Euler-Mascheroni constant digits
|
Euler product
|
Hadjicostas’s formula
|
jeep problem
|
Mertens theorem
|
Stieltjes constants
Related Wolfram Language symbol
EulerGamma
Educational grade level
college level
Subject classifications
Show details
MathWorld
Euler-Mascheroni constant
|
unsolved problems
Associated person
Leonhard Euler
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MRB constant
Assuming "MRB constant" is referring to a mathematical definition | Use as
a mathematical constant
instead
Input interpretation
MRB constant
Illustration
Definition
Fewer details
Consider the sequence of partial sums defined by
s
n

n
∑
k1
k
(-1)
1/k
k
.
As can be seen in the plot above, the sequence has two limit points at
-0.812140…
and 0.187859… (which are separated by exactly 1). The upper limit point is sometimes known as the MRB constant after the initials of its original investigator (Burns 1999; Plouffe).
Sums for the MRB constant are given by
S
=
lim
N∞
2N
∑
n1
n
(-1)
1/n
n
=
1+
lim
N∞
2N+1
∑
n1
n
(-1)
1/n
n
=
∞
∑
k1
[
1/(2k)
(2k)
-
1/(2k-1)
(2k-1)
]
=
∞
∑
k1
k
(-1)
(
1/k
k
-1)
=
0.1878596424…
(Finch 2003, p. 450; OEIS A037077).
Related terms
Glaisher-Kinkelin constant
|
power tower
|
Steiner’s problem
Subject classifications
Show details
MathWorld
miscellaneous constants