User xbh

45 votes

Accepted

How can I justify this without determining the determinant?

answered Nov 24, 2019 at 18:17

13 votes

Accepted

Value of $\lim_{n\to \infty} \frac{1}{2^n} \sum_{k=1}^{2^n} \{\log_{2}k\}$

answered Sep 4, 2018 at 5:11

11 votes

Accepted

Solve: $\log_3(5(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})+2^{64})=?$

answered Sep 8, 2018 at 17:25

10 votes

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$ \begin{vmatrix} f(a) & g(a) & h(a) \\ f(b) & g(b) & h(b) \\ f'(c) & g'(c) & h'(c) \\ \end{vmatrix} =0 $

answered Oct 24, 2018 at 15:16

9 votes

Accepted

find the value of $\int_{-a}^{a} \frac{f(x)} {1+e^x} dx $?

answered Nov 13, 2018 at 1:30

9 votes

Justify: if $x\gt 0$, $\;\lim_{n\to\infty} \sqrt{n}\cdot{\overbrace{\sin\sin\cdots\sin}^{n\space\text{sines}}(x)}=\sqrt{3}$

answered Jul 30, 2018 at 16:13

8 votes

Definite integral of $x\sin^n x$ from $0$ to $\pi/2$

answered Sep 15, 2018 at 13:16

8 votes

Accepted

How to simplify or upperbound this summation?

answered Sep 18, 2018 at 16:45

7 votes

Prove $\forall x \in \mathbb{R}$: $[\sinh(x)+\cosh(x)]^n = \cosh(nx)+\sinh(nx)$ ; $ n\in \mathbb{Q}$

answered Aug 6, 2018 at 4:20

7 votes

Evaluate $\lim_{n \to \infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{\frac{1}{n}}$

answered Aug 13, 2018 at 16:58

7 votes

Accepted

If $I$ an interval and $x_1,x_2,x_3\in I$, why $t_1x_1+t_2x_2+t_3x_3\in I$ when $t_1+t_2+t_3=1, t_i\in [0,1]$?

answered Aug 21, 2018 at 10:09

7 votes

Accepted

Evaluate $ \lim\limits_{x \to 0}\frac{\int_0^{x^2}f(t){\rm d}t}{x^2\int_0^x f(t){\rm d}t}.$

answered Jan 12, 2019 at 3:49

6 votes

What is the main difference between pointwise and uniform convergence as defined here?

answered Jan 6, 2019 at 4:51

6 votes

Limit for $e$ and $\frac{1}{e}$

answered Apr 14, 2019 at 16:45

6 votes

How do you integrate $\int_{0}^{\infty}\frac{a\cos{(cx)}}{a^2+x^2}dx$?

answered Jun 14, 2019 at 2:44

6 votes

Accepted

Solve the following equation: $\sqrt {\sin x - \sqrt {\cos x + \sin x} } = \cos x$

answered Jun 30, 2019 at 4:55

6 votes

Limit $ \lim_{n \rightarrow \infty}\sqrt{(1+\frac{1}{n}+\frac{1}{n^3})^{5n+1}} $

answered Dec 5, 2018 at 12:21

6 votes

Accepted

$\int_0^{\infty}\frac{\sin x}{(1+x)^2}\,dx$ converges absolutely

answered Dec 12, 2018 at 6:37

6 votes

Easy ways to calculate matrix exponential of a particular $4\times 4$ matrix

answered Aug 5, 2018 at 13:42

6 votes

Accepted

Does $\sum^{\infty}_{n=1}xe^{-nx}$ converge uniformly on $[0,\infty)$?

answered Sep 14, 2018 at 14:19

5 votes

Accepted

Does $\lim_{n\to\infty} \sum^{n^2}_{k=n}\frac{1}{k}$ exist?

answered Sep 6, 2018 at 14:19

5 votes

Accepted

Suppose $f(x)$ is differentiable on $[0,1]$, and $f(0)=0$, $f(x)\ne 0,\forall x\in(0,1)$

answered Aug 22, 2018 at 5:22

5 votes

Accepted

$f$ is differentiable in $[0,1]$ ,$\sup_{x\in[0,1]}|f'(x)| \le M\lt+\infty $

answered Aug 15, 2018 at 3:06

5 votes

Accepted

Convergence of $\sum_{n=1}^{\infty} \sin\left(\frac{\pi}{2}n\right)\left(n(-1)^n+\left(\sin\frac{1}{n}\right)^{-1}\right)$

answered Aug 7, 2018 at 10:45

5 votes

Does $(1+\frac12-\frac13) + (\frac14+\frac15-\frac16)+(\frac17+\frac18-\frac19)+\cdots$ converge?

answered Jan 6, 2019 at 3:13

5 votes

Suppose $ \alpha, \beta>0 $. Compute: $ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $

answered Oct 22, 2018 at 14:15

5 votes

Accepted

Help find the mistake in this problem of finding limit (using L'Hopital)

answered Oct 24, 2018 at 14:31

5 votes

Permute columns by pre-multiplying and rows by post-multiplying?

answered Feb 14, 2019 at 3:53

5 votes

Determinant of matrix $[a_{ij}]$ where $a_{ij}=ij$ if $i\ne j$ and $a_{ij}=1+ij$ if $i=j$

answered Sep 23, 2019 at 18:06

4 votes

Accepted

$\int _{2}^{+\infty }\ \dfrac{\sin x}{x\ln x }dx$ is convergent but not absolutely convergent

answered Apr 4, 2022 at 4:30

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313 Answers

How can I justify this without determining the determinant?

Value of $\lim_{n\to \infty} \frac{1}{2^n} \sum_{k=1}^{2^n} \{\log_{2}k\}$

Solve: $\log_3(5(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})+2^{64})=?$

$ \begin{vmatrix} f(a) & g(a) & h(a) \\ f(b) & g(b) & h(b) \\ f'(c) & g'(c) & h'(c) \\ \end{vmatrix} =0 $

find the value of $\int_{-a}^{a} \frac{f(x)} {1+e^x} dx $?

Justify: if $x\gt 0$, $\;\lim_{n\to\infty} \sqrt{n}\cdot{\overbrace{\sin\sin\cdots\sin}^{n\space\text{sines}}(x)}=\sqrt{3}$

Definite integral of $x\sin^n x$ from $0$ to $\pi/2$

How to simplify or upperbound this summation?

Prove $\forall x \in \mathbb{R}$: $[\sinh(x)+\cosh(x)]^n = \cosh(nx)+\sinh(nx)$ ; $ n\in \mathbb{Q}$

Evaluate $\lim_{n \to \infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{\frac{1}{n}}$

If $I$ an interval and $x_1,x_2,x_3\in I$, why $t_1x_1+t_2x_2+t_3x_3\in I$ when $t_1+t_2+t_3=1, t_i\in [0,1]$?

Evaluate $ \lim\limits_{x \to 0}\frac{\int_0^{x^2}f(t){\rm d}t}{x^2\int_0^x f(t){\rm d}t}.$

What is the main difference between pointwise and uniform convergence as defined here?

Limit for $e$ and $\frac{1}{e}$

How do you integrate $\int_{0}^{\infty}\frac{a\cos{(cx)}}{a^2+x^2}dx$?

Solve the following equation: $\sqrt {\sin x - \sqrt {\cos x + \sin x} } = \cos x$

Limit $ \lim_{n \rightarrow \infty}\sqrt{(1+\frac{1}{n}+\frac{1}{n^3})^{5n+1}} $

$\int_0^{\infty}\frac{\sin x}{(1+x)^2}\,dx$ converges absolutely

Easy ways to calculate matrix exponential of a particular $4\times 4$ matrix

Does $\sum^{\infty}_{n=1}xe^{-nx}$ converge uniformly on $[0,\infty)$?

Does $\lim_{n\to\infty} \sum^{n^2}_{k=n}\frac{1}{k}$ exist?

Suppose $f(x)$ is differentiable on $[0,1]$, and $f(0)=0$, $f(x)\ne 0,\forall x\in(0,1)$

$f$ is differentiable in $[0,1]$ ,$\sup_{x\in[0,1]}|f'(x)| \le M\lt+\infty $

Convergence of $\sum_{n=1}^{\infty} \sin\left(\frac{\pi}{2}n\right)\left(n(-1)^n+\left(\sin\frac{1}{n}\right)^{-1}\right)$

Does $(1+\frac12-\frac13) + (\frac14+\frac15-\frac16)+(\frac17+\frac18-\frac19)+\cdots$ converge?

Suppose $ \alpha, \beta>0 $. Compute: $ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $

Help find the mistake in this problem of finding limit (using L'Hopital)

Permute columns by pre-multiplying and rows by post-multiplying?

Determinant of matrix $[a_{ij}]$ where $a_{ij}=ij$ if $i\ne j$ and $a_{ij}=1+ij$ if $i=j$

$\int _{2}^{+\infty }\ \dfrac{\sin x}{x\ln x }dx$ is convergent but not absolutely convergent