# Tagged Questions

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### $-\varepsilon\log(x)\overset{?}{\geq} -\log(\varepsilon x)$

I'm refering to this proof: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result In there it's stated that "Since the matrix $(P_{ij})_{ij}$ is a doubly stochastic matrix and $-\log$ is a ...
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### A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*)$$ while evaluating this log-cosine integral. I made several ...
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### A series with only rational terms for $\ln \ln 2$

We all know that $$\ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}.$$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
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### Noncircular construction of $e$ and $\ln$ for the real line

Could anyone direct me to (or possibly detail) a construction of $e$ and $\ln$ along the reals? For example, they can define $e=\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$ but from this definition ...
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### Trying to prove a function is increasing in one variable after the second variable crosses a particular value.

Let $f(p,n)$ = $\ln\bigl(1 + \frac{ap}{b+I}\bigr) + \sum_{j = 1}^{n}\ln\bigl(1+ \frac{b_j}{b + I - b_j + ap}\bigr)$, where $b>0$ and $a>0$ are fixed constants. The positive real numbers $b_j$ ...
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### Proof $\lim\limits_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$

I want to show that for all $a \in \mathbb{R }$ $$\lim_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$$ So far i've got $\lim\limits_{n \rightarrow \infty}ne^{(\frac{1}{n}\log a)}-n$, but when i ...
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