#harmonicnumber — Public Fediverse posts
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Euler–Mascheroni constant! :euler:
In fact, the last one is:
\[\large\displaystyle\int_1^{+\infty}\mathrm dx\ \left(\frac{1}{\lfloor x\rfloor}-\frac1x\right)=\gamma\approx0.5772156649\]Equivalently,
\[\large\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^n \frac1{k}-\ln n\right)=\gamma=0.5772156649\ldots\]
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Unsolved problem in mathematics:
Is Euler–Mascheroni constant irrational? If so, is it transcendental?#Euler #Mascheroni #EulerMascheroni #Constant #gamma #EulerConstant #EulersConstant #EulerMascheroniConstant #Irrational #Irrationality #Transcendental #Transcendence #Unsolved #UnsolvedProblem #Maths #Mathematics #Indeterminate #IndeterminateForm #IndeterminateForms #Inf #Infinity #HarmonicNumber #HarmonicNumbers #HarmonicSeries #Logarithm #Log #Logarithms #NaturalLogarithm #Integral #ImproperIntegral
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Try to prove the following two results that relate the harmonic numbers to the golden ratio. Have an excellent weekend.
\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^n}=2\sqrt5\ln\varphi\]
\[\displaystyle\sum_{n=1}^\infty\binom{2n}n\dfrac{H_n}{5^nn}=\frac{2\pi^2}{15}-2\ln^2\varphi\]
where \(\varphi=\frac{1+\sqrt5}2\) is the golden ratio; and \(H_n=\left(1+\frac12+\frac13+\ldots+\frac1n\right)\) is the \(n\)-th harmonic number.
#GoldenRatio #HarmonicNumbers #HarmonicNumber #Logarithm #Pi #Summation #Math #Sum #InfiniteSum #Binomial #BinomialCoefficient #Maths #WeekendChallenge