User gegu

5 votes

3 answers

2k views

why $\Bbb Z_2\times\Bbb Z_2$ and $\Bbb Z_4$ are not isomorphic?

asked Apr 9, 2017 at 1:47

5 votes

2 answers

785 views

(Revised) Prove that $\phi$ is a group homomorphism and find the kernel.

asked Apr 16, 2017 at 1:19

5 votes

2 answers

9k views

Prove that If $f$ is integrable on $[a,b]$ and $[b,c]$ then $f$ is integrable on $[a,c]$

asked Jul 31, 2017 at 4:34

5 votes

1 answer

2k views

Evaluate Riemann integral $\int_{a}^{b} e^x dx$ using upper and lower integral definitions and theorems

asked Jul 28, 2017 at 3:46

5 votes

4 answers

209 views

Show that $ \left( 1 + \frac{x}{n}\right)^n$ is uniformly convergent on $S=[0,1]$. [duplicate]

asked Aug 14, 2017 at 0:11

5 votes

1 answer

2k views

If $h(r, \theta) = f(r \cos \theta, r \sin \theta)$, show that $ f_{xx}+f_{yy} = h_{rr} + \frac{1}{r} h_r + \frac{1}{r^2} h_{00}$

asked Aug 21, 2017 at 9:00

4 votes

2 answers

87 views

Prove, if $h(u,v)=f(\sin u + \cos v)$ then $h_u \sin v +h_v \cos u = 0.$

asked Aug 21, 2017 at 3:22

4 votes

3 answers

2k views

Given that $f(x) = -1$ if $f$ is irrational and $f(x)=1$ if $f$ is rational, show that $f$ is not continuous anywhere.

asked Jul 11, 2017 at 22:16

4 votes

1 answer

18k views

Show that f is discontinuous at every rational and continuous at every irrational. [duplicate]

asked Jul 12, 2017 at 0:53

3 votes

2 answers

11k views

Cosets of S3 and Permutations

asked Apr 5, 2017 at 23:48

3 votes

2 answers

5k views

If G is an abelian group and $n \in \Bbb N$, show that $\phi :G \rightarrow G$ defined by $ g \mapsto g^n$ is a group homomorphism

asked Apr 18, 2017 at 1:45

3 votes

3 answers

78 views

From $a_{n+1}=(1+ \frac{1}{n})^n \cdot a_n$ to $a_n$ to be proven by induction

asked May 25, 2017 at 3:56

3 votes

4 answers

709 views

Show that $\sum_{m=0}^n(-1)^m\binom nm=0$

asked May 27, 2017 at 6:35

3 votes

2 answers

7k views

Prove by induction that $ \sin(x) +\sin(3x) +...+ \sin [(2n-1)x]= \frac{1-\cos(2nx)}{2\sin x}$

asked May 27, 2017 at 12:30

3 votes

2 answers

2k views

Show that $ \sqrt{p}$ is irrational if $p$ is prime

asked Jun 3, 2017 at 23:06

3 votes

2 answers

519 views

First four nonzero terms of the McLaurin expansion of $\frac{xe^x}{\sin x}$ at $x_0=0$

asked Aug 17, 2017 at 11:40

3 votes

4 answers

883 views

Find the limit , Show that $\{F_n\}$ converges uniformly to $F$ on closed subsets of $S$, but not on $S$. $F_n(x) = x^n \sin nx, S=(-1,1)$

asked Aug 13, 2017 at 2:09

3 votes

0 answers

285 views

Given $\lim\limits_{x \rightarrow x_0} \frac{f(x)-p(x)}{(x-x_0)^n} =0$ Show that $a_r= \frac{f^{(r)}(x_0)}{r!}$

asked Jul 18, 2017 at 5:26

3 votes

0 answers

106 views

Prove that $f$ is not integrable over $[a,b]$

asked Jul 26, 2017 at 16:41

2 votes

1 answer

839 views

Prove the Riemann integral $\int_{a}^{b} x^2 dx = \frac{b^3-a^3}{3}$ by using the mean value theorem

asked Jul 27, 2017 at 3:43

2 votes

3 answers

418 views

Determine convergence or divergence of $ \int_0^{\infty} \frac{1 + \cos^2x}{\sqrt{1+x^2}} dx$

asked Aug 1, 2017 at 23:28

2 votes

2 answers

149 views

Bounding $\int_0^{\infty} x^p e^{-x} dx$ to find the values of p for integral convergence.

asked Aug 2, 2017 at 6:14

2 votes

2 answers

49 views

From $\lim_{t\to0^+}\frac{t^2\cos\phi\sin\phi\sin(t\cos\phi)}{t^3}$ to $\cos^2\phi\sin\phi$

asked Aug 20, 2017 at 7:09

2 votes

1 answer

54 views

For $g(X) = \frac{(x^2+y^4)^3}{1+x^6y^4}$ , show $\lim\limits_{|x| \rightarrow \infty} g(x,ax) = \infty? $ for any real number $a$

asked Aug 19, 2017 at 5:00

2 votes

3 answers

92 views

Rewriting $a+b + |a- b|$

asked May 18, 2017 at 3:05

2 votes

1 answer

651 views

Proving that $\sup(S+T)=\sup(S)+\sup(T)$

asked May 19, 2017 at 7:28

2 votes

2 answers

106 views

kernel, group, and subgroup...

asked Apr 14, 2017 at 21:00

2 votes

2 answers

5k views

Prove that $\mathbb{C}^*$ is isomorphic to a subgroup of $GL_2(\mathbb{R})$

asked Apr 10, 2017 at 2:57

2 votes

3 answers

2k views

Prove or disprove that $U(8) \cong Z_4$

asked Apr 12, 2017 at 0:33

2 votes

0 answers

39 views

Describe $S=\{x\mid x\in\mathbb Z\}$ as open, closed, or neither, and find $S^o$, $(S^c)^o$, and $(S^o)^c$

asked Jun 13, 2017 at 5:41

Stack Exchange Network

109 Questions

why $\Bbb Z_2\times\Bbb Z_2$ and $\Bbb Z_4$ are not isomorphic?

(Revised) Prove that $\phi$ is a group homomorphism and find the kernel.

Prove that If $f$ is integrable on $[a,b]$ and $[b,c]$ then $f$ is integrable on $[a,c]$

Evaluate Riemann integral $\int_{a}^{b} e^x dx$ using upper and lower integral definitions and theorems

Show that $ \left( 1 + \frac{x}{n}\right)^n$ is uniformly convergent on $S=[0,1]$. [duplicate]

If $h(r, \theta) = f(r \cos \theta, r \sin \theta)$, show that $ f_{xx}+f_{yy} = h_{rr} + \frac{1}{r} h_r + \frac{1}{r^2} h_{00}$

Prove, if $h(u,v)=f(\sin u + \cos v)$ then $h_u \sin v +h_v \cos u = 0.$

Given that $f(x) = -1$ if $f$ is irrational and $f(x)=1$ if $f$ is rational, show that $f$ is not continuous anywhere.

Show that f is discontinuous at every rational and continuous at every irrational. [duplicate]

Cosets of S3 and Permutations

If G is an abelian group and $n \in \Bbb N$, show that $\phi :G \rightarrow G$ defined by $ g \mapsto g^n$ is a group homomorphism

From $a_{n+1}=(1+ \frac{1}{n})^n \cdot a_n$ to $a_n$ to be proven by induction

Show that $\sum_{m=0}^n(-1)^m\binom nm=0$

Prove by induction that $ \sin(x) +\sin(3x) +...+ \sin [(2n-1)x]= \frac{1-\cos(2nx)}{2\sin x}$

Show that $ \sqrt{p}$ is irrational if $p$ is prime

First four nonzero terms of the McLaurin expansion of $\frac{xe^x}{\sin x}$ at $x_0=0$

Find the limit , Show that $\{F_n\}$ converges uniformly to $F$ on closed subsets of $S$, but not on $S$. $F_n(x) = x^n \sin nx, S=(-1,1)$

Given $\lim\limits_{x \rightarrow x_0} \frac{f(x)-p(x)}{(x-x_0)^n} =0$ Show that $a_r= \frac{f^{(r)}(x_0)}{r!}$

Prove that $f$ is not integrable over $[a,b]$

Prove the Riemann integral $\int_{a}^{b} x^2 dx = \frac{b^3-a^3}{3}$ by using the mean value theorem

Determine convergence or divergence of $ \int_0^{\infty} \frac{1 + \cos^2x}{\sqrt{1+x^2}} dx$

Bounding $\int_0^{\infty} x^p e^{-x} dx$ to find the values of p for integral convergence.

From $\lim_{t\to0^+}\frac{t^2\cos\phi\sin\phi\sin(t\cos\phi)}{t^3}$ to $\cos^2\phi\sin\phi$

For $g(X) = \frac{(x^2+y^4)^3}{1+x^6y^4}$ , show $\lim\limits_{|x| \rightarrow \infty} g(x,ax) = \infty? $ for any real number $a$

Rewriting $a+b + |a- b|$

Proving that $\sup(S+T)=\sup(S)+\sup(T)$

kernel, group, and subgroup...

Prove that $\mathbb{C}^*$ is isomorphic to a subgroup of $GL_2(\mathbb{R})$

Prove or disprove that $U(8) \cong Z_4$

Describe $S=\{x\mid x\in\mathbb Z\}$ as open, closed, or neither, and find $S^o$, $(S^c)^o$, and $(S^o)^c$