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.