query: How to prove that $\pi > e\ln(\pi)$?
positive_passages.docid: 21003
positive_passages.text: Comparing $\pi^e$ and $e^\pi$ without calculating them
negative_passages.docid: 122115
negative_passages.text: Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
negative_passages.docid: 62417
negative_passages.text: Prove This Inequality ${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$
negative_passages.docid: 26608
negative_passages.text: How to show that $\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3?$
negative_passages.docid: 65562
negative_passages.text: Proving the inequality $\arctan\frac{\pi}{2}\ge1$
negative_passages.docid: 33392
negative_passages.text: Proving that $ \ln\left(\frac{49}{50}\right)<\sum^{98}_{k=1}\int^{k+1}_{k}\frac{k+1}{x(x+1)}dx<\ln(99)$
negative_passages.docid: 48881
negative_passages.text: Verification the integral identity $\int_0^\pi \ln(1+\alpha\cos(x))dx=\pi\ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$.
negative_passages.docid: 23946
negative_passages.text: Prove the definitions of $e$ to be equivalent
negative_passages.docid: 6449
negative_passages.text: Prove logarithmic inequality with greatest integer function.
negative_passages.docid: 54638
negative_passages.text: Integral $\int^{ \pi /2}_{0} \ln (\sin x)\ dx$
negative_passages.docid: 39943
negative_passages.text: Proving $\mathrm e <3$
negative_passages.docid: 95266
negative_passages.text: Show $|t|\leq \pi \Rightarrow |e^{it}-1|\geq 2|t|/ \pi$
negative_passages.docid: 124209
negative_passages.text: How is $\ln(-1) = i\pi$?
negative_passages.docid: 4287
negative_passages.text: Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$
negative_passages.docid: 90796
negative_passages.text: Inequality Regarding e
negative_passages.docid: 64571
negative_passages.text: Prove that $\int_{E}f =\lim \int_{E}f_{n}$
negative_passages.docid: 25797
negative_passages.text: Integral of $ x \ln( \sin (x))$ from 0 to $ \pi $