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 ...
127
votes
10answers
28k views
Multiple-choice question about the probability of a random answer to itself being correct
I found this math "problem" on the internet, and I'm wondering if it has an answer:
Question: If you choose an answer to this question at random, what is the probability that you will be correct?
...
43
votes
12answers
2k views
Is it morally right and pedagogically right to google answers to homework? [closed]
This is a soft question that I have been struggling with lately.
My professor sets tough questions for homework (around 10 per week).
The difficulty is such that if I attempt the questions entirely ...
39
votes
4answers
1k views
Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even
I have a question:
Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then prove that
$$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$
How can I ...
35
votes
7answers
2k views
The last digit of $2^{2006}$
My 13 year old son was asked this question in a maths challenge. He correctly guessed 4 on the assumption that the answer was likely to be the last digit of $2^6$. However is there a better ...
31
votes
6answers
2k views
Is $2^{218!} +1$ prime?
Prove that $2^{218!} +1$ is not prime.
I can prove that the last digit of this number is $7$, and that's all.
Thank you.
30
votes
7answers
2k views
How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement
$$\lim\limits_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my ...
28
votes
4answers
2k views
Finding the limit of $\frac {n}{\sqrt[n]{n!}}$
I'm trying to find
$$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$
I tried couple of methods: Stolz, Squeeze, D'Alambert
Thanks!
Edit: I can't use Stirling.
27
votes
2answers
389 views
Very curious properties of ordered partitions relating to Fibonacci numbers
I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon.
We call an ordered ...
27
votes
0answers
1k views
Simplicial homology of real projective space by Mayer-Vietoris
Consider the $n$-sphere $S^n$ and the real projective space $\mathbb{RP}^n$. There is a universal covering map $p: S^n \to \mathbb{RP}^n$, and it's clear that it's the coequaliser of $\mathrm{id}: S^n ...
25
votes
6answers
941 views
What are some good math specific study habits?
What are/ were some of your good mathematician's study habits that you found really worked for you? I'm a CS major at a respected school and have a solid GPA... However, I definitely lack when it ...
22
votes
3answers
1k views
Norms on C[0, 1] inducing the same topology as the sup norm
This is an old homework problem of mine that I was never able to solve. The solution may or may not involve the Baire category theorem, which I am terrible at applying.
Let $C[0, 1]$ denote the ...
22
votes
2answers
432 views
When is a sum of consecutive squares equal to a square?
We have the sum of squares of $n$ consecutive positive integers: $$S=(a+1)^2+(a+2)^2+ ... +(a+n)^2$$ Problem was to find the smallest $n$ such, that $S=b^2$ will be square of some positive integer. I ...
22
votes
2answers
431 views
How to prove that $\frac{(5m)!(5n)!}{(m!)(n!)(3m+n)!(3n+m)!}$ is a natural number?
How to prove that $$\frac{(5m)! \cdot (5n)!}{m! \cdot n! \cdot (3m+n)! \cdot (3n+m)!}$$ is a natural number $\forall m,n\in\mathbb N$ , $m\geqslant 1$ and $n\geqslant 1$?
Thanks in advance.
21
votes
1answer
402 views
Convergence of the series $\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$.
Please determine whether the series $\displaystyle\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$ converges.
(Note: In Mathematica, the result tends to converge. Moreover, this is a problem mis-copied from ...
19
votes
3answers
9k views
Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$
Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$.
This question is just after the definition of differentiation and the theorem that if $f$ is finitely ...
18
votes
6answers
1k views
Helping my daughter with her homework: solving an algebra word problem.
Three bags of apples and two bags of oranges weigh $32$ pounds.
Four bags of apples and three bags of oranges weigh $44$ pounds.
All bags of apples weigh the same. All bags of oranges weigh the ...
18
votes
5answers
985 views
Approximation to $ \sqrt{2}$
I'm a first year Undergraduate student from India. Our professor is going to start a Real Analysis course in September and I was preparing for the initials. I tried and solved many problems, but this ...
18
votes
2answers
469 views
Surprising but simple group theory result on conjugacy classes
I have read that for any group $G$ of order $2m+1$ (odd) with $n$ conjugacy classes, it is always the case that $16$ divides the value $(2m+1)-n = |G|-n$.
This seems to me like an astonishing ...
18
votes
4answers
336 views
Limit of series involving ratio of two factorials
$$
\sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3}
$$
The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
18
votes
2answers
1k views
Is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$
I need a hint. The problem is: is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$
I'm pretty sure that there aren't any, but so far I couldn't find the proof.
My best idea so far is ...
18
votes
3answers
487 views
If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$
I'm trying to prove the following:
If $(a_n)$ is a sequence of positive numbers such that $\sum_{n=1}^\infty a_n b_n<\infty$ for all sequences of positive numbers $(b_n)$ such that ...
18
votes
2answers
669 views
Tricky contour integral
I am trying to evaluate the definite integral
$$\int_0^\infty \frac{\sin ax\ dx}{x^2+b^2}$$
where $a,b>0$. This is a problem on an assignment for a class in complex variables. I understand the ...
18
votes
1answer
321 views
Easy proof, that $\rm e\notin \mathbb Q$
$\def\e{{\rm e}}$
I recently had the task to explain the proof that $\e$ is irrational as a presentation to my classmates. To prepare this presentation, the teacher gave me a script with a proof that ...
18
votes
1answer
739 views
$\sum_{k=0}^{100} a_k x^{k}=0, a_i \in \mathbb{Z}$ How many equations?
Let $x^{100}+a_{99} x^{99}+a_{98} x^{98}+ \dots +a_{1}x +a_{0}=0, \ \ a_{100}=1, a_i \in \mathbb{Z}$, is an algebraic equation with integer coefficients.
Assume that all (100 with multiplicity) ...
18
votes
0answers
390 views
Computing the Chern-Simons invariant of SO(3)
I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
17
votes
5answers
2k views
Integral of $\frac{1}{(1+x^2)^2}$
I am in the middle of a problem and having trouble integrating the following integral:
$$\int_{-1}^1\frac1{(1+x^2)^2}\mathrm dx$$
I tried doing partial fractions and got:
$$1=A(1+x^2)+B(1+x^2)$$
I ...
16
votes
5answers
4k views
If $a^2$ divides $b^2$, then $a$ divides $b$
Let $a$ and $b$ be positive integers. Prove that: If $a^2$ divides $b^2$, then $a$ divides $b$.
Context: the lecturer wrote this up in my notes without proving it, but I can't seem to figure out ...
16
votes
5answers
456 views
How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer?
$\forall m,n\in\mathbb Z$ , $m\ge1$ and $n\ge1$ how to prove that $$\frac{(mn)!}{m!(n!)^m}$$ is an integer?
Thanks in advance.
16
votes
3answers
194 views
How to prove summation, multiplication, subtraction of two roots of $1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0$ aren't rationals?
Assume $a$, $b$ are distinct roots of the following equation: $$1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0,$$ where $p$ is a prime number and $p \gt 2$. How to prove that $ab$, $a+b$, $a-b$ are not ...
16
votes
2answers
347 views
Prove $\det(\mathbf I+\mathbf A^T\mathbf A) = \det (\mathbf I+\mathbf A\mathbf A^T)$
Given a matrix $\mathbf A\in \mathbb R^{m\times n}$ and an identity matrix $\mathbf I$ of appropriate dimensions, how do you prove $\det(\mathbf I+\mathbf A^T\mathbf A) = \det (\mathbf I+\mathbf ...
