User Jason

3 votes

1 answer

235 views

$n$ identical boxes , $3$ different rooms in which they can be put. Each room can hold a maximum of 24 boxes.(do not have to use all the rooms)

asked Sep 4, 2011 at 16:35

1 vote

3 answers

171 views

$(f(c)-f(a))(f(b)-f(c))<0$ prove that there exists a $d$ such that $f'(d)=0$.

asked Jan 13, 2012 at 10:14

6 votes

6 answers

8k views

Proving that $e^x>1+x^2$

asked Jan 25, 2012 at 9:56

9 votes

5 answers

9k views

If Monty Hall doesn't know where the prize is, should the contestant still switch doors, after Monty opens one door and unveiss a goat?

asked May 28, 2011 at 11:42

3 votes

1 answer

302 views

Are there quicker and less error prone methods for finding determinant? [duplicate]

asked Jul 25, 2018 at 17:51

1 vote

2 answers

489 views

Show that a subspace $U$ exists such that the orthogonal projection of $v$ onto it has the specified length $\alpha < \|v\|$

asked Aug 4, 2018 at 16:03

3 votes

3 answers

2k views

Finding a Particular Coefficient Using Generating Functions

asked Oct 6, 2011 at 15:10

1 vote

1 answer

395 views

CFG Grammar for $L=\left\{ a^ib^jc^k\mid i+k=3j \right\}$

asked Jul 13, 2019 at 11:15

1 vote

0 answers

66 views

If $f(n)=\omega(g(n))$ then exists $h(n)$ so $h(n)=\omega(g(n))$ and $f(n)=\omega(h(n))$?

asked Feb 13, 2019 at 13:44

0 votes

2 answers

27 views

Given $T(v)=v-a\left<v,w_1\right>w_1-a\left<v,w_2\right>w_2$ where $a\in \mathbb R$ and $||w_1||=||w_2||=1, w_1\perp w_2$ Show that $T=T^*$

asked Aug 16, 2018 at 8:08

1 vote

1 answer

95 views

Finding the Jordan Form of a transformation defined by $T(X)=AX$ when $A,X \in M_{4\times 4}(\mathbb C)$

asked Aug 14, 2018 at 11:31

1 vote

1 answer

111 views

Given square real matrix $A$ with $\det(A) = 108$ and $(A-2I)(A^2-9I)=0$, is $A$ normal?

asked Aug 13, 2018 at 8:56

1 vote

1 answer

177 views

Given $A,B$ self adjoint matrices in $M_{n\times n}(\mathbb C)$ with positive real eigenvalues. Show that the eigenvalues of $AB$ are positive.

asked Aug 10, 2018 at 13:57

0 votes

1 answer

42 views

Given $f(x)$ such that $f'(0)=2;f(0)=1$ use the maclaurin expansion of $f(x)$ to calculate $\lim_{x\to 0}(f(x))^{1/x}$.

asked Jul 4, 2018 at 14:17

0 votes

0 answers

69 views

If $a_n$ is a decreasing series of positive numbers show that $c_n = \sqrt[n]{\prod_{i=1}^n a_i}+ \sqrt[n]{\sum_{i=1}^n a_i^n}$ converges

asked Jul 4, 2018 at 8:23

2 votes

2 answers

126 views

Does the following series converge uniformly in $[1,\infty)$? $\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)\cdots (1+x^n)}$

asked Jul 1, 2018 at 12:43

2 votes

2 answers

85 views

If $\int_a^\infty f(x) dx$ converges, does there exist $0<c<1$ such that for $c \le p \le 1,$ $\int_a^\infty f^p(x)$ converges?

asked Jun 30, 2018 at 17:32

0 votes

2 answers

63 views

Calculate $\lim_{n\to\infty} \frac{\sum_{i=1}^n ia_i}{n^2+n}$ given that $a_n \to a$

asked Jun 25, 2018 at 8:51

1 vote

2 answers

140 views

Does the infinite series $\sum_{n=2} \frac{(-1)^n}{\sqrt[n]{ln(n)}}$ converge absolutely / converge / diverge?

asked May 17, 2018 at 16:59

3 votes

1 answer

1k views

Given $A_{m\times n}$ and $B_{n \times m} (m<n)$. prove that AB is not singular and BA is singular

asked Jul 9, 2014 at 8:36

4 votes

5 answers

84 views

If $x_0$ is a real root of $p(x)=x^4+a_3 x^3 + a_2 x^2 +a_1 x + a_0$ and $p'(x_0) \ne 0$. Does $p(x)$ have at least two real roots?

asked Apr 26, 2018 at 8:17

0 votes

3 answers

56 views

If $a_n \le b_n \le a_{n+1}$ prove that $\overline{\lim}_{n \to \infty} a_n=\overline{\lim}_{n \to \infty} b_n$

asked Apr 25, 2018 at 8:17

1 vote

3 answers

98 views

If $f(x)$ is differentiable in $\mathbb{R}$ & $a,b \in \mathbb{R}, a\neq b$ such that $f'(x)=(x-a)(x-b)$ then $f$ has exactly one local min and max?

asked Apr 24, 2018 at 15:50

7 votes

2 answers

135 views

Prove or contradict: Between each two solutions of $\arctan x = \sin x$ exists a solution for $1-\cos x = x^2 \cos x$

asked Apr 24, 2018 at 12:41

1 vote

3 answers

105 views

Calculate the following series limit: $\lim_{n \to \infty} (\frac{n}{n+3})^\sqrt{n(1+n)}$

asked Apr 24, 2018 at 10:31

1 vote

2 answers

157 views

If $a_n > 0$ and $\lim_{n \to \infty}{\sqrt{a_n}}>1$ show that $\lim_{n \to \infty}a_n$ exists.

asked Apr 4, 2018 at 8:51

3 votes

2 answers

184 views

Is there a more convenient method for converting a base to be orthogonal than Gram Schmidt?

asked Feb 16, 2018 at 13:55

0 votes

1 answer

34 views

If $V$ over $\mathbb{C}$, $\dim V\ge 2$, let $q: V \rightarrow \Bbb{C}$ be a Quadratic form. s.t. exists $v \ne 0$ such that $q(v)=0$ [duplicate]

asked Feb 15, 2018 at 12:24

0 votes

1 answer

28 views

What is the dimension of the solution vector space for $M=-M^t$ ($M\in M_{n\times n}^{\mathbb{C}})$?

asked Feb 13, 2018 at 11:38

-1 votes

3 answers

532 views

Prove that if $0<a<b<\pi/2$ then $\tan^{-1}b-\tan^{-1} a<\tan b -\tan a$

asked Feb 11, 2018 at 13:18

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81 Questions

$n$ identical boxes , $3$ different rooms in which they can be put. Each room can hold a maximum of 24 boxes.(do not have to use all the rooms)

$(f(c)-f(a))(f(b)-f(c))<0$ prove that there exists a $d$ such that $f'(d)=0$.

Proving that $e^x>1+x^2$

If Monty Hall doesn't know where the prize is, should the contestant still switch doors, after Monty opens one door and unveiss a goat?

Are there quicker and less error prone methods for finding determinant? [duplicate]

Show that a subspace $U$ exists such that the orthogonal projection of $v$ onto it has the specified length $\alpha < \|v\|$

Finding a Particular Coefficient Using Generating Functions

CFG Grammar for $L=\left\{ a^ib^jc^k\mid i+k=3j \right\}$

If $f(n)=\omega(g(n))$ then exists $h(n)$ so $h(n)=\omega(g(n))$ and $f(n)=\omega(h(n))$?

Given $T(v)=v-a\left<v,w_1\right>w_1-a\left<v,w_2\right>w_2$ where $a\in \mathbb R$ and $||w_1||=||w_2||=1, w_1\perp w_2$ Show that $T=T^*$

Finding the Jordan Form of a transformation defined by $T(X)=AX$ when $A,X \in M_{4\times 4}(\mathbb C)$

Given square real matrix $A$ with $\det(A) = 108$ and $(A-2I)(A^2-9I)=0$, is $A$ normal?

Given $A,B$ self adjoint matrices in $M_{n\times n}(\mathbb C)$ with positive real eigenvalues. Show that the eigenvalues of $AB$ are positive.

Given $f(x)$ such that $f'(0)=2;f(0)=1$ use the maclaurin expansion of $f(x)$ to calculate $\lim_{x\to 0}(f(x))^{1/x}$.

If $a_n$ is a decreasing series of positive numbers show that $c_n = \sqrt[n]{\prod_{i=1}^n a_i}+ \sqrt[n]{\sum_{i=1}^n a_i^n}$ converges

Does the following series converge uniformly in $[1,\infty)$? $\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)\cdots (1+x^n)}$

If $\int_a^\infty f(x) dx$ converges, does there exist $0<c<1$ such that for $c \le p \le 1,$ $\int_a^\infty f^p(x)$ converges?

Calculate $\lim_{n\to\infty} \frac{\sum_{i=1}^n ia_i}{n^2+n}$ given that $a_n \to a$

Does the infinite series $\sum_{n=2} \frac{(-1)^n}{\sqrt[n]{ln(n)}}$ converge absolutely / converge / diverge?

Given $A_{m\times n}$ and $B_{n \times m} (m<n)$. prove that AB is not singular and BA is singular

If $x_0$ is a real root of $p(x)=x^4+a_3 x^3 + a_2 x^2 +a_1 x + a_0$ and $p'(x_0) \ne 0$. Does $p(x)$ have at least two real roots?

If $a_n \le b_n \le a_{n+1}$ prove that $\overline{\lim}_{n \to \infty} a_n=\overline{\lim}_{n \to \infty} b_n$

If $f(x)$ is differentiable in $\mathbb{R}$ & $a,b \in \mathbb{R}, a\neq b$ such that $f'(x)=(x-a)(x-b)$ then $f$ has exactly one local min and max?

Prove or contradict: Between each two solutions of $\arctan x = \sin x$ exists a solution for $1-\cos x = x^2 \cos x$

Calculate the following series limit: $\lim_{n \to \infty} (\frac{n}{n+3})^\sqrt{n(1+n)}$

If $a_n > 0$ and $\lim_{n \to \infty}{\sqrt{a_n}}>1$ show that $\lim_{n \to \infty}a_n$ exists.

Is there a more convenient method for converting a base to be orthogonal than Gram Schmidt?

If $V$ over $\mathbb{C}$, $\dim V\ge 2$, let $q: V \rightarrow \Bbb{C}$ be a Quadratic form. s.t. exists $v \ne 0$ such that $q(v)=0$ [duplicate]

What is the dimension of the solution vector space for $M=-M^t$ ($M\in M_{n\times n}^{\mathbb{C}})$?

Prove that if $0<a<b<\pi/2$ then $\tan^{-1}b-\tan^{-1} a<\tan b -\tan a$