Difference between revisions of "Euler-Mascheroni constant"
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===Alternate formulation of the limit=== | ===Alternate formulation of the limit=== | ||
− | The tangent-line approximation (first-degree [[Taylor polynomial]]) of <math>\ln(k + 1)</math> about <math>x = k</math> is <cmath>\ln(k + 1) = \ln(k) + ((k + 1) - k)\ln'(k) + E_k</cmath> for some error term <math>E_k</math>. Using <math>\ln'(x) = \frac{1}{x}</math> and simplifying, <cmath>\ln(k + 1) = \ln(k) + \frac{1}{k} + E_k.</cmath> Applying the tangent-line formula [[Recursion|recursively]] for all <math>k</math> descending from <math>n</math> to <math>1</math>, | + | The tangent-line approximation (first-degree [[Taylor polynomial]]) of <math>\ln(k + 1)</math> about <math>x = k</math> is <cmath>\ln(k + 1) = \ln(k) + ((k + 1) - k)\ln'(k) + E_k</cmath> for some error term <math>E_k</math>. Using <math>\ln'(x) = \frac{1}{x}</math> and simplifying, <cmath>\ln(k + 1) = \ln(k) + \frac{1}{k} + E_k.</cmath> Applying the tangent-line formula [[Recursion|recursively]] for all <math>k</math> descending from <math>n - 1</math> to <math>1</math>, |
<cmath> | <cmath> | ||
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Hence, <math>\gamma = - \sum_{k=1}^{\infty} E_k</math> is a defined constant. | Hence, <math>\gamma = - \sum_{k=1}^{\infty} E_k</math> is a defined constant. | ||
+ | |||
+ | ==See also== | ||
+ | *[[Harmonic series]] | ||
+ | *[[Natural logarithm]] | ||
[[Category: Constants]] | [[Category: Constants]] |
Latest revision as of 16:32, 19 September 2022
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.