query: How can I prove without using a calculator that $\frac{1}{e} > \frac{\ln \pi}{\pi}$?
positive_passages.docid: 21003
positive_passages.text: Comparing $\pi^e$ and $e^\pi$ without calculating them
negative_passages.docid: 124026
negative_passages.text: Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$
negative_passages.docid: 130078
negative_passages.text: How can one prove that $\pi^4 + \pi^5 < e^6$?
negative_passages.docid: 61659
negative_passages.text: How to show that $ \sum_{n = 0}^{\infty} \frac {1}{n!} = e$?
negative_passages.docid: 70715
negative_passages.text: How do I prove $\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx = n\pi$?
negative_passages.docid: 122971
negative_passages.text: how to prove that $\ln(1+x)< x$
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
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: 30897
negative_passages.text: $\pi(x)\geqslant\frac{\log x}{2\log2}$ for all $x\geqslant2.$
negative_passages.docid: 28141
negative_passages.text: How to know if $\log_78 > \log_89$ without using a calculator?
negative_passages.docid: 127759
negative_passages.text: Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$
negative_passages.docid: 98838
negative_passages.text: How to evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$
negative_passages.docid: 14013
negative_passages.text: How to prove that $\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2)... 2n} = \frac{4}{e}$
negative_passages.docid: 27195
negative_passages.text: Evaluating $\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$
negative_passages.docid: 102334
negative_passages.text: How can you prove $\frac{n(n+1)(2n+1)}{6}+(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$ without much effort?
negative_passages.docid: 48225
negative_passages.text: Proving that $\sin1 $(radian) is irrational without using Taylor Series Expansion.
negative_passages.docid: 34905
negative_passages.text: Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$