Prove $\frac{1}{n+1}\le \log\left(1+\frac{1}{n}\right)\le \frac{1}{n} ,\forall n\ge1$ if $\log x= \int_{1}^{x}\frac{dt}{t},x>0$
43226
Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$
145909
Prove that $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+....+\frac{1}{2n}\geq \frac{2}{3}$
28669
Find limit of the following sequence: $\frac{\log(n+1)}{\log(n)}$?
138516
$\sum (-1)^{n+1}\log\left(1+\frac{1}{n}\right)$ convergent but not absolutely convergent
147977
Evaluate $\int_0^1\left(\frac{x-1}{x+1}\right)^n\frac{1}{\ln x} \,dx$
64800
Evaluate $\int_0^1 \log \left( \frac{x^2+\sqrt{3}x+1}{x^2-\sqrt{3}x+1} \right) \frac{dx}{x} $
104447
Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$
3110
Prove $\frac{1}{\sqrt{x}}\geq \frac{\ln x}{x-1}$
30688
Prove that $\frac{x_1}{1+x_2+x_3+\ldots+x_n}+\frac{x_2}{1+x_1+x_3+\ldots+x_n}+\ldots+\frac{x_n}{1+x_1+x_2+\ldots+x_{n-1}}\ge\frac{n}{2n-1}$.
22575
Prove that $\lim\limits_{n\rightarrow\infty} \left(1+\frac{1}{a_{n}} \right)^{a_{n}}=e$ if $\lim\limits_{n\rightarrow\infty} a_{n}=\infty$
25921
Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing
48503
Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
67622
Proving of Integral $\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx = \ln\left(\frac{a}{b}\right)$
97253
How prove $e^x|f(x)|\le 2$ if $f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$
47790
If $x \geq 2$ prove that $\sum_{ n \leq x} \frac{1}{\phi(n)} = \mathcal{O}(\log(x))$
57686
Finding $ \int^1_0 \frac{\ln(1+x)}{x}dx$
79466
Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$