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 ...
11
votes
3answers
286 views
Collatz-ish Olympiad Problem
The following is an Olympiad Competition question, so I expect it to have a pretty solution:
For a positive integer $d$, define the sequence:
\begin{align}
a_0 &= 1\\
a_n &=
...
11
votes
2answers
1k views
On Lipschitz condition and absolute continuity
A function $f(x)$ on $[0,1]$ is said to satisfy a Lipschitz condition if there exists a constant $M$, such that $$|f(x)-f(y)|\leqslant M|x-y| ~\forall~x,y\in[0,1].
$$
I want to show the ...
11
votes
2answers
373 views
Finding $\lim_{x \to 0}\ \frac{\sin(\cos(x))}{\sec(x)}$
The problem is to find:
$\lim_{x \to 0}\ \dfrac{\sin(\cos(x))}{\sec(x)}$
I rewrite the equation as follows:
$\lim_{x \to 0}\ \dfrac{\sin(\cos(x))}{\dfrac{1}{\cos(x)}}$
And multiply by ...
11
votes
5answers
341 views
Prove that the sequence$ c_1 = 1$, $c_{n+1} = 4/(1 + 5c_n) $ , $ n \geq 1$ is convergent and find its limit
Prove that the sequence $c_{1} = 1$, $c_{(n+1)}= 4/(1 + 5c_{n})$ , $n \geq 1$ is convergent and find its limit.
Ok so up to now I've worked out a couple of things.
$c_1 = 1$
$c_2 = 2/3$
$c_3 = ...
11
votes
2answers
108 views
Prove that in a ring with $x^3 = x$, $x+x+x+x+x+x=0$.
This was an excercise on a course on abstract algebra at the University of Groningen. I have been working on this for ages, but I can't seem to figure it out.
Problem
Let $R$ be a ring with $\forall ...
11
votes
1answer
140 views
Integer Sequence “sums of digits of squares”.
For all $n \in \mathbb{N}$ we define the function $\delta(n)=p$, where $p$ is sums of digits of $n^2$. For example if $n=17, \ n^2=289$, then $\delta(17)=2+8+9=19$.
Let $a_k$ is a monotonically ...
11
votes
1answer
401 views
Exercise 6.5 in Humphrey's Book on Lie Algebras
I am trying to solve Exercise 6.5 part 4 in James Humphreys' Introduction to Lie Algebras and Representation Theory. I added the (homework) tag because my question is about an exercise, but this is ...
11
votes
1answer
511 views
Converse of the Weierstrass $M$-Test?
I was assigned a few problems in my Honors Calculus II class, and one of them was kind of interesting to do:
Suppose that $f_{n}$ are nonnegative bounded functions on $A$ and let $M_{n} = \sup ...
11
votes
1answer
206 views
Convergence of the series $\sum_{n=1}^\infty \frac{(2n)!!}{(2n+1)!!} $
Study the convergence of the next series: $$\sum_{n=1}^\infty \frac{(2n)!!}{(2n+1)!!} $$
My solution: since $$\frac{(2n)!!}{(2n+2)!!} \leq \frac{(2n)!!}{(2n+1)!!}$$
forall $n \in \mathbb{N}$ and ...
11
votes
1answer
869 views
Is there error in the answer to Spivak's Calculus, problem 5-3(iv)?
I'm puzzled by the answer to a problem for Spivak's Calculus (4E) provided in his Combined Answer Book.
Problem 5-3(iv) (p. 108) asks the reader to prove that $\mathop{\lim}\limits_{x \to a} x^{4} ...
11
votes
0answers
579 views
Spivak's “Differential Geometry” Volume 1, Chapter 1 ,Problem #20 part (b)
Problem 20 part (b) of Chapter 1 asks us to show that the infinite-holed torus is homeomorphic to the "infinite jail cell window." His hint helped me to get started (I think).
(I apologize for not ...
10
votes
11answers
2k views
Solving $5^n > 4,000,000$ without a calculator
If $n$ is an integer and $5^n > 4,000,000.$ What is the least possible value of $n$? (answer: $10$)
How could I find the value of $n$ without using a calculator ?
10
votes
5answers
485 views
Is it true that $\lvert \sin z \rvert \leq 1$ for all $z\in \mathbb{C}$?
Is it true that $\left\lvert \sin z \right \rvert \leq 1$ for all $z \in \mathbb{C}$ ?
I think that is not true, can anyone help me?
10
votes
10answers
1k views
Proving $\sqrt 3$ is irrational.
There is a very simple proof by means of divisivility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum:
Suppose
...
10
votes
6answers
588 views
Given $a_{1}=1, \ a_{n+1}=a_{n}+\frac{1}{a_{n}}$, find $\lim \limits_{n\to\infty}\frac{a_{n}}{n}$
I started by showing that $1\leq a_{n} \leq n$ (by induction) and then $\frac{1}{n}\leq \frac{a_{n}}{n} \leq 1$ which doesn't really get me anywhere.
On a different path I showed that $a_{n} \to ...
10
votes
4answers
785 views
Help with summing a power series
I'd like to determine the function corresponding to the following power series:
$$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1},
$$
where ...
10
votes
5answers
308 views
A matrix satisfying $AB-BA=B$
If $A$ and $B$ are two matrices of $\mathcal{M}_n(\mathbb{R}$) such that
$$AB-BA=B$$
how can we prove that $B$ isn't invertible?
my attempt: I found that $\mathrm{tr}(B)=0$ but I know that this is ...
10
votes
3answers
248 views
Is $\mathbb{Q}/\mathbb{Z}$ isomorphic to $\mathbb{Q}$?
Is $\mathbb{Q}/\mathbb{Z}$ isomorphic to $\mathbb{Q}$?
My guess is no. Does the first isomorphism theorem have anything to do with this?
Any hints appreciated, thanks.
10
votes
5answers
421 views
How to calculate $\int_{-a}^{a} \sqrt{a^2-x^2}\ln(\sqrt{a^2-x^2})\mathrm{dx}$
Well,this is a homework problem.
I need to calculate the differential entropy of random variable
$X\sim f(x)=\sqrt{a^2-x^2},\quad -a<x<a$ and $0$ otherwise. Just how to calculate
$$
...
10
votes
4answers
599 views
I want to know why $\omega \neq \omega+1$.
In Kunen's book, Set Theory,chapter I.7, he said: $1+\omega=\omega \neq \omega+1$.
I want to know why $\omega \neq \omega+1$.
10
votes
7answers
246 views
Check if $\det(I + S) = 1 + \operatorname{trace}(S)$ holds ?
I saw the following statement in my homework and we are asked to make use of the statement:
If $S$ is a symmetric matrix then
$$\det(I + S ) = 1 + \operatorname{trace}(S).$$
However, I am not ...
10
votes
2answers
334 views
If $(a_{n})$ is increasing, is $u_{n}=\frac{a_{1}+\cdots+a_{n}}{n}$ increasing as well?
And what about the other direction? If $(u_{n})$ is increasing, what about $(a_{n})$?
I'm guessing the former is true since we know that if $(a_{n})$ converges, $(u_{n})$ converges to the same limit. ...
10
votes
4answers
402 views
Showing $f'(x) = f(x)$ implies an exponential function [duplicate]
Possible Duplicate:
Proof that $\exp(x)$ is the only function for which $f(x) = f'(x)$
How can I show the statement $f'(x) = f(x)$ implies the function is defined as $f: \mathbb{R} ...
10
votes
6answers
462 views
Complex roots of $z^3 + \bar{z} = 0$
I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre.
Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
10
votes
5answers
1k views
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem ...
10
votes
2answers
190 views
If $f\in\mathbb{Q}[X]$ and $f(\mathbb{Q})=\mathbb{Q}$ then $\deg f=1$
Could you give me any hint how to prove that if $f\in\mathbb{Q}[X]$ and $f(\mathbb{Q})=\mathbb{Q}$ then degree of $f$ is equal to $1$? I have tried to prove it by using Bézout's theorem but I guess ...
10
votes
3answers
765 views
Stirling number of the first kind: Proof of Recursion formula
I want to prove this recursion formula for Stirling numbers of the first kind:
$$s_{n+1,k+1} = \sum_{i=k}^{n} \binom{i}{k} s_{n,i}$$
But I lack a useful idea. Perhaps someone could inspire me?
...
10
votes
4answers
555 views
Proving that if $f'$ has at most $n-1$ zeros, then $f$ has at most $n$ zeros
Is this proof correct?
The problem is the following.
Let $n$ be a natural number. Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ is differentiable and that the following equation has at ...
10
votes
3answers
539 views
What is so special about negative numbers $m$, $\mathbb{Z}[\sqrt{m}]$?
This question is based on a homework exercise:
"Let $m$ be a negative, square-free integer with at least two prime factors. Show that $\mathbb{Z}[\sqrt{m}]$ is not a PID."
In an aside comment in the ...
10
votes
2answers
398 views
Show $f(x) = x^3 - \sin^2{x} \tan{x} < 0$ on $(0, \frac{\pi}{2})$
This is the last of a homework problem set from Principles of Mathematical Analysis (Ch. 8 #18(a)) that I've been working/stuck on for a few days:
Define $f(x) = x^3 - \sin^2{x}\tan{x}.$
Find ...
10
votes
2answers
149 views
Prove the identity $ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$
$$ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$$
Class themes are: Generating functions and formal power series.
10
votes
2answers
217 views
coequalizers+pullbacks implies equalizers?
The question is on the title, I would like a hint on this exercise.
This is what I've tried so far:
Suppose we're given $f,g:A\rightarrow B$, let $h=\operatorname{Coeq}(f,g)$, then we have parallel ...
10
votes
2answers
673 views
Proving that the tensor product is right exact
Let
$A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$
any ring).
I am trying to prove that ...
10
votes
2answers
310 views
Summation by parts of $\sum_{k=0}^{n}k^{2}2^{k}$
I want to evaluate this sum
$$\sum_{k=0}^{n}k^{2}2^{k}$$ by summation by parts (two times) and I need to know, if my approach was right.
I know the formula for summation by parts is $$\sum u\Delta ...
10
votes
1answer
469 views
On integrals related to $\int^{+\infty}_{-\infty} e^{-x^2} dx = \sqrt{\pi}$
You are given the result that $$\int^{+\infty}_{-\infty} e^{-x^2} dx = \sqrt{\pi}$$
a. Use this result to find $$\int^{+\infty}_{-\infty} e^{-ax^2} dx$$
b. Use the above results to find ...
10
votes
2answers
365 views
A bound on the Fourier coefficients of an $\alpha$-Lipschitz function
I am asked to show that if $0 < \alpha < 1$, and if $f \in \Lambda^\alpha(\mathbb{T})$, then we have for $k\neq 0$, $$|\widehat{f}(k)| \leq \pi^\alpha \frac{\|f\|_{\Lambda^1}}{k^\alpha}$$
I ...
10
votes
2answers
118 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 ...
10
votes
3answers
193 views
A trigonometric series
Let $\alpha$ be a real number. I'm asked to discuss the convergence of the series
$$
\sum_{k=1}^{\infty} \frac{\sin{(kx)}}{k^\alpha}
$$
where $x \in [0,2\pi]$.
Well, I show you what I've done:
if ...
10
votes
3answers
325 views
Proving the number of permutations $A,B\;$ with $n+1$ total cycles and $AB=(123\cdots n)$ is $C_n$
Please give a combinatorial proof of the following:
The number of pairs $(A, B)$ of permutations of the set $\{ 1, 2,\ldots,n \}$ such that they have a total of $n+1$ cycles and their composition ...
10
votes
3answers
347 views
On a combinatorial identity
I'm trying to prove that if we have the elementary symmetric polynomials that the following identity holds:(where $x = (x_1,..,x_n)$ is a vector of n variables)
$$\sum_{k=0}^n e_k(x)^2 = x_1\cdots x_n ...
10
votes
2answers
456 views
If $|H|$ and $[G:K]$ are relatively prime, then $H \leq K$
Let $H$ and $K$ be subgroups of a finite group $G$, at least one of which is normal. Show that if $|H|$ and $[G:K]$ are relatively prime, then $H \leq K$.
In the case that $K$ is normal, let $\pi : G ...
10
votes
1answer
99 views
Functoriality of the Fundamental group
The fundamental group is a functor from the category of pointed topological spaces to the category of groups.
Therefore every base-point preserving continuous function $f$ between pointed ...
10
votes
1answer
139 views
Calculate Asymptotics of Integral?
Let $f$ be a continuous function on $[0,1]$. How do I calculate the asymptotics, as $n\rightarrow\infty$, of
$\displaystyle \int_{[0,1]^n}f\left(\frac{x_1+...+x_n}{n}\right)\text d x_1...\text d ...
10
votes
1answer
84 views
Show that $x^4+1$ is reducible in p-adic numbers $\mathbb{Q}_p$ for p>2 prime.
This is a homework problem for algebraic number theory but I'm having trouble getting started. Do I use induction in general, or show this holds for $p \equiv 1,3$ (mod 4)?
Any help would be ...
10
votes
1answer
233 views
$(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$
I did the following homework question, can you tell me if I have it right?
We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
10
votes
1answer
405 views
Four kissing circles
How can one go about solving the following problem?
Inscribe a circle in an arbitrary triangle. Call it's radius $r_1$.
Inscribe three more circles so that each one is tangent to two sides of
...
10
votes
1answer
122 views
Compute $\chi(\mathbb{C}\mathrm{P}^2)$.
I need to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using techniques from differential topology. I cannot think of any theorems that are particularly useful for this computation, so I think that I will ...
10
votes
1answer
197 views
Words of Length $n$ over the Alphabet $\{1,2,3\}$ with Certain Restrictions
Let $w(n)$ denote the number of words of length $n$ over the alphabet $\{1,2,3\}$ with the restrictions that in a word the parity of $1$s be even and the parity of $2$s odd.
I have written out the ...
10
votes
1answer
366 views
Prove that minimum of $\lambda \sin \theta + (1 - \lambda) \cos \theta \le -\dfrac{1}{\sqrt 2}$
I need a little nudge to the finish for the last bit of this problem.
Express $\lambda \sin \theta + (1 - \lambda) \cos \theta$ in the form $R \sin (\theta + \phi)$, where $R(R>0)$ and $\tan ...
10
votes
4answers
148 views
Galois Theory and Galois Groups
Show that $\mathbb{Q}[x]/\langle x^{3}-2\rangle = [{a + b\alpha + c\alpha^{2}: a, b, c \in \mathbb{Q}, \alpha^{3} = 2}]$ is not a Galois extension of $\mathbb{Q}$. In particular, show that every ...
