query: If $y=x^{n-1}\ln (x)$, then prove that $xy_{n}=(n-1)!$
positive_passages.docid: 155096
positive_passages.text: If $f(x)=x^{n-1}\log x$, then the $n$-th derivative of $f$ is equals?
negative_passages.docid: 27580
negative_passages.text: Use mathematical induction to prove $ (2n)!\geq 2^n(n!)^2$ for $n \in \mathbb{N}$
negative_passages.docid: 30835
negative_passages.text: Logarithm proof problem: $a^{\log_b c} = c^{\log_b a}$
negative_passages.docid: 108429
negative_passages.text: Show that if $x_n \to x$ then $\sqrt{x_n} \to \sqrt{x}$
negative_passages.docid: 93408
negative_passages.text: Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$
negative_passages.docid: 19804
negative_passages.text: How to prove that $|w_{n+1}-w_n| \leq (2/3)^2 |w_n - w_{n-1}|$?
negative_passages.docid: 76267
negative_passages.text: Prove that: $x_1\cdot x_2\cdots x_n>y_1\cdot y_2\cdots y_m$.
negative_passages.docid: 136077
negative_passages.text: Prove that $x-1$ is a factor of $x^n-1$
negative_passages.docid: 82775
negative_passages.text: if $a>1$, Prove that $\lim a^{1\over n}=1$
negative_passages.docid: 115799
negative_passages.text: Proving $x^{\log n} = n^{\log x}$
negative_passages.docid: 122929
negative_passages.text: Prove that $Y=-2\sum_{i=1}^n \ln F_{X_i}(X_i)$ have distribution $\chi^2(2n)$
negative_passages.docid: 140409
negative_passages.text: Prove that $\sqrt{n} > \ln n$
negative_passages.docid: 16774
negative_passages.text: If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.
negative_passages.docid: 88995
negative_passages.text: Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
negative_passages.docid: 82742
negative_passages.text: Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $
negative_passages.docid: 86535
negative_passages.text: Prove quotient of two $N(0,1)$ is $\text{Cauchy}(0,1)$
negative_passages.docid: 49408
negative_passages.text: Proof of the derivative of ln(x)