Euler-Mascheroni constant
The Euler-Mascheroni constant is a constant defined by the limit
Its value is approximately
Whether is rational or irrational and (if irrational) algebraic or transcendental is an open question.
Contents
[hide]Proof of existence
Alternate formulation of the limit
The tangent-line approximation (first-degree Taylor polynomial) of about
is
for some error term
. Using
and simplifying,
Applying the tangent-line formula recursively for all
descending from
to
,
Because , we may rearrange to
Adding
to both sides yields
Taking the limit as
goes to infinity of both sides,
Thus, .
Convergence of the sum of error terms
We have . For
, the maximum absolute value of
for
is
. Therefore, by the Lagrange Error Bound,
The series famously converges to
by the Basel problem, so
converges to
and
converges to
.
Because for all
, the Series Comparison Test gives that
must converge to a value in
.
Hence, is a defined constant.