Answer by Jack D'Aurizio for Closed form and/or approximation for...
By exploiting Frullani's theorem$$ \log(k) = \int_{0}^{+\infty}\frac{e^{-x}-e^{-kx}}{x}\,dx \tag{1}$$we get that:$$ f_n =...
View ArticleAnswer by Did for Closed form and/or approximation for...
There is no hope to get exact formulas (since, for example, the logarithms of the prime numbers are linearly independent on the rational numbers). However, one can get rather precise asymptotics...
View ArticleAnswer by Claude Leibovici for Closed form and/or approximation for...
As comments already said, I doubt that a closed form could exist.However, in terms of approximation, considering the function $$f_n=\sum_{i=1}^{n-1}\binom{n}{i}\, i\,\log(i)$$ is is interesting to...
View ArticleClosed form and/or approximation for...
Is there any closed form for the expression : $$f_n=\sum_{i=1}^{n-1}\binom{n}{i}\cdot i\cdot \log(i)\ ?$$If not, how to get an approximation?
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