Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...
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1answer
168 views
Learning mathematics
Now im not sure this is the appropriate place to ask this so please forgive me if its not.
i have only recently discovered an interested in mathematics and well i could only take a year off work to be ...
10
votes
1answer
611 views
Minkowski Inequality for $p \le 1$
I've been trying to prove the concavity of a particular function which I reduced to proving the reverse Minkowski Inequality for $p \le 1$, $p \ne 0$ for arguments in $\mathbb{R}^{n}_{+}$. That is,
...
9
votes
6answers
594 views
Proof of the equality $\sum\limits_{k=1}^{\infty} \frac{k^2}{2^k} = 6$
Show that for $k$ running over positive integers
$$ \sum_{k=1}^\infty \frac{k^2}{2^k}=6 .$$ We can use finite calculus.
9
votes
6answers
991 views
Show that the set of all finite subsets of $\mathbb{N}$ is countable.
Show that the set of all finite subsets of $\mathbb{N}$ is countable.
I'm not sure how to do this problem. I keep trying to think of an explicit formula for 1-1 correspondence like adding all the ...
9
votes
2answers
396 views
How to evaluate the series $1+\frac34+\frac{3\cdot5}{4\cdot8}+\frac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots$
How can I evaluate the following series:
$$1+\frac{3}{4}+\frac{3\cdot 5}{4\cdot 8}+\frac{3\cdot 5\cdot 7}{4\cdot 8\cdot 12}+\frac{3\cdot 5\cdot 7\cdot 9}{4\cdot 8\cdot 12\cdot 16}+\cdots$$
In one ...
9
votes
4answers
402 views
How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?
I can show that the following limit exists but
I am having difficulties to find it. It is
$$\lim_{n\to \infty} \sum_{k=1}^n \frac{k^n}{n^n}$$
Can someone please help me?
9
votes
6answers
224 views
Can I compare a series beginning with $n=0$ to one that begins with $n=1$?
I'm doing a question for homework, and I am required to use the Comparison Test to test for convergence.
The series in question is:
$$ \sum_{n=0}^\infty \frac {n-1}{{(n+2)}^3} $$
The series I would ...
9
votes
5answers
296 views
Determine the number of solutions of the equation $n^m = m^n$ [duplicate]
Possible Duplicate:
$x^y = y^x$ for integers $x$ and $y$
Determine the number of solutions of the equation
$n^m = m^n$
where both m and n are integers.
9
votes
4answers
493 views
Find $\cos(x+y)$ if $\sin(x)+\sin(y)= a$ and $\cos(x)+\cos(y)= b$
Find $\cos(x+y)$ if $\sin(x)+\sin(y)= a$ and $\cos(x)+\cos(y)= b$.
9
votes
6answers
1k views
Why is a finite integral domain always field?
This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse:
let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$,
then as $R$ is closed under ...
9
votes
2answers
263 views
Classifying Unital Commutative Rings of Order $p^2$
I'm trying to classify unital commutative rings of order $p^2$, where $p$ is a prime. At first, I happened to neglect the 'unital' and 'commutative' requirements, and after an arduous route I managed ...
9
votes
4answers
280 views
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
Suppose that $f(x)$ is continuous on $(0, +\infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function. My attempt is to show that for any point $a \neq b$ , we have ...
9
votes
2answers
272 views
Ranges and the Fundamental Theorem of Calculus 1
I'm going over a chapter by chapter review for my calculus final and discovered this problem:
$$y=\int_{\sqrt{x}}^{x^3}\sqrt{t}\sin{t}\;\mathrm dt$$
They split it up so that it became:
...
9
votes
5answers
702 views
Help solving $\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}dx}$
$\displaystyle\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}\,\mathrm{d}x}$
I used partial fractions, solved $A = 2, C = 3$.
$$\frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} ...
9
votes
2answers
299 views
Show $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \|f\|_{\infty}$ for $f \in L^{\infty}$
I have a question that I need help with getting started (possibly I would be back for more help).
I have a measure space $(X,A,\mu)$ that is finite, and $f \in L^{\infty}(\mu)$. Also, defined is ...
9
votes
2answers
447 views
$\Delta x$ in limit problem?
I was working on some limit homework and everything was going fine until I reached this problem:
$$\lim_{\Delta x \to 0} \frac{2(x + \Delta x) - 2x}{\Delta x}.$$
I am understanding limits but the ...
9
votes
2answers
313 views
Gaussian Primes
I need to adapt the Sieve of Eratosthenes for the usual integers to find all Gaussian primes with norm less than a specific limit. How to apply it to finding all Gaussian primes with norm less than ...
9
votes
3answers
219 views
Value of $\lim_{n\to\infty}{(1+\frac{2n^2+\cos{n}}{n^3+n})^n}$
How should one go about computing $$\lim_{n\to\infty}{\left(1+\frac{2n^2+\cos{n}}{n^3+n}\right)^n}\quad?$$ What surprised me about this is that
...
9
votes
2answers
122 views
Unusual evaluation of $\sum \frac{1}{n^2}$
Assume the formula $$\sum_{n=-\infty}^\infty\frac{1}{(n+u)^2}=\frac{\pi^2}{(\sin \pi u)^2},$$ where $u\notin\Bbb Z$. I have been trying to prove that ...
9
votes
3answers
177 views
Prove that $(e+x)^{e-x}>(e-x)^{e+x}$
I get stuck with proving that $$(e+x)^{e-x}>(e-x)^{e+x}$$ for $x \in (0, e)$. All I know, is that it is doable with Jensen inequality, and I started with defining $$f(x)=(e+x)^{e-x}$$ and further ...
9
votes
2answers
226 views
For which $p \in \mathbb{R}_{>0}$ does the integral $\int_{[0,1]^n} \frac{\mathrm dx}{(x_1^p+2x_2^p + … + nx_n^p)^{1/3}}$ converge?
I want to find out for which $p \in \mathbb{R}_{>0}$ the integral
$$\int_{[0,1]^n} \frac{\mathrm d x}{(x_1^p+2x_2^p + ... + nx_n^p)^{1/3}}$$
converges. To be honest, I have no idea or whatsoever ...
9
votes
3answers
442 views
choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries?
choose a random number between 0 and 1 and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds 1. What's the expected ...
9
votes
2answers
660 views
How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
I want to prove the following theorem (no idea whether it has a name):
Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...
9
votes
3answers
170 views
Integral calculation help
I have this integral
$$\int_0^{\infty}{\frac{e^{-ax}-e^{-bx}}{x}\sin{mx} \, dx} \quad (a > 0 \, , b >0)$$
What I did was this
$$ \begin{align}
...
9
votes
2answers
115 views
Group of invertible elements
Let R be a ring with unity. How can I prove that group of invertible elements of R is never of order 5? My teacher told me and my colleagues that problem is very hard to solve. I would be glad if ...
9
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2answers
256 views
9
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2answers
116 views
The existence of a group automorphism with some properties implies commutativity.
Let $G $ be a finite group, $T$ be an automorphisom of $ G $ st $ Tx = x \iff x=e $. Suppose further that $ T^2 =I $. Prove that $ G $ is abelian.
I was thinking if I show $ T aba^{-1} b^ ...
9
votes
1answer
137 views
Is the Euler characteristic $\chi =2$ for the prism with a hole?
I keep getting $\chi=2$ for the solid in the picture. It's a prism with a hole joining two opposite sides. I remember reading that $\chi=0$ for such solids.
Help me find my error. I'd appreciate if ...
9
votes
1answer
180 views
Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$
Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$
converges?
I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
9
votes
1answer
119 views
I want to prove that the following set is closed
Let $A\subseteq R$ be a compact set and $B\subseteq R$ closed. Then $S=\{b\sin a;b\in B,a\in A\}$ is closed.
What I have done is to consider the continuous function $$f:\mathbb{R}\times ...
9
votes
3answers
299 views
Triangle problem
I have got one simple task to prove:
We have got a triangle $\triangle XYZ$.
Then we create points $A,B,C$ on $XY, YZ, ZX$ respectively, such that $XA = AB = BZ$ and $CZ = AY = AC$.
How to prove ...
9
votes
1answer
475 views
What does a zero tensor product imply?
I'm trying to prove that for two finitely generated $A$-modules $M,N$ ($A$ being any ring), the tensor product $M\otimes_A N$ is zero iff $Ann(M)+Ann(N)=A$.
The if direction is of course easy- just ...
9
votes
1answer
248 views
Question about the converse of a well known result from Linear Algebra
I am a graduate student studying for a Linear Algebra qualifying exam and I have been going over sample problems from previous exams. The recommended text for these problems are Hoffman and Kunze ...
9
votes
2answers
288 views
Zeros of Fourier transform of a function in $C[-1,1]$
I am trying to prove the following:
Let $g \in C[-1,1]$. Then the function $$G(z) = \int_{-1}^1 e^{itz}g(t)dt$$ has infinitely many zeros.
I know that $G(z)$ is entire and $\lim_{x \to \pm ...
9
votes
1answer
216 views
Why study schemes?
Why study schemes instead of only affine/projective varieties, given by zeros of polynomials in the affine/projective space? I mean, what is gained by introducing the concept of schemes?
Thank you!
9
votes
1answer
152 views
Evaluating series with factorial denominator (sanity check).
Is my approach to evaluating this series correct?
$$\sum_{n=1}^\infty \frac{n}{(n+1)!}$$
Has partial sum equivalent to:
$$S_m = \sum_{n=1}^m \frac{n}{(n+1)!} = \sum_{j=2}^{m+1} \frac{j-1}{j!} = ...
9
votes
3answers
242 views
A question on a compact space
Show:
If the closure of every discrete subset of a space is compact then the whole space is compact.
Thanks advance:)
9
votes
1answer
262 views
Relation between congruence subgroups. $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$
I'm hoping there's a pleasant way to solve this one.
Prove that $\Gamma(M)\Gamma(N) = \Gamma(\gcd(M,N))$.
Showing that $\Gamma(M)\Gamma(N) \subset \Gamma(\gcd(M,N))$ is rather straight forward, ...
9
votes
2answers
213 views
Let $f:[a,b]\to\mathbb R$ be Riemann integrable and $f>0$. Prove that $\int_a^bf>0$. (Without Measure theory)
The suggestion above is not relevent to my question.
I've been struggling with this for a while, and I have a couple of leads that kind of got me nowhere:
At first I thought that if $f$ is ...
9
votes
2answers
223 views
Check my workings: Prove the limit $\lim\limits_{x\to -2} (3x^2+4x-2)=2 $ using the $\epsilon,\delta$ definition.
Prove the limit $\lim\limits_{x\to -2} (3x^2+4x-2)=2 $ using the $\epsilon,\delta$ definition.
Precalculations
My goal is to show that for all $\epsilon >0$, there exist a $\delta > 0$, ...
9
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3answers
812 views
how to prove DEF is an equilateral triangle?
ABC is an equilateral triangle，and AD = BE = CF，Prove DEF is an equilateral triangle.
9
votes
1answer
181 views
Class equation of subgroup of $SL(4,\mathbb{F}_2)$
Can you point me toward a computation-light derivation of the class equation of the subgroup of $SL(4,\mathbb{F}_2)$ consisting of upper-triangular matrices with 1's on the main diagonal?
The ...
9
votes
1answer
333 views
Limit of a decreasing sequence of outer measures is an outer measure?
The problem given to me on my homework is:
Prove that the limit of a decreasing family of outer measures is an outer measure.
Doing out the "obvious" approach, we quickly reach the problem of ...
9
votes
2answers
347 views
Exact power of $p$ that divides the discriminant of an algebraic number field
I am doing Marcus problem 21 (b) of chapter 3. The setup for this problem is given in problem 20:
Setup: Let $L/K$ be a finite extension of algebraic number fields. Write $R = \mathcal{O}_K$ ...
9
votes
1answer
91 views
Showing that a CW space is contractible if it is endowed with a certain binary operation
I am having trouble with the following homework problem, and was hoping someone could provide me with a hint: I am given a connected CW space $X$ which has a continuous associative operation $(x,\ ...
9
votes
1answer
642 views
how to prove a bounded linear operator is compact?
I encountered a homework problem that says: If A is a bounded linear operator from X to Y. And K is a compact operator from X to Y. Here X and Y are both Banach. And $R(A)\subset R(K)$. I need to show ...
9
votes
1answer
84 views
Ideal in compact Hausdorff space
This is exercise 70, chapter 4. from Folland (page 142)
Let $X$ be a compact Hausdorff space. An ideal in $C(X, \mathbb{R})$ is a subalgebra $J$ of
$C(X, \mathbb{R})$ such that if $f\in J$ and $g\in ...
9
votes
1answer
189 views
Compactness Theorem Application
I am doing an exercise on the compactness theorem of first order logic. The task is to prove that there is no singe first order sentence which is satisfied in exactly the infinite graphs (thereby, a ...
9
votes
3answers
301 views
Help me prove $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$
Please help me prove this Leibniz equation: $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$. Thanks!
9
votes
2answers
146 views
How to prove that a group with some properties is abelian?
Let $(G,.)$ be a group and $m,n\in\mathbb Z$ such that $\gcd(m,n)=1 $ and
$$ \forall a,b \in G:a^mb^m=b^ma^m$$
$$\forall a,b \in G:a^nb^n=b^na^n.$$ Then how prove $G$ is an abelian group ?
Thanks ...
