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 ...
17
votes
8answers
1k views
How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$
How can if find the sum for:
$$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$
I know the answer thanks to Wolfram Alpha. I'm more concerned with how to get to to that number. It cites tests to prove ...
131
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?
...
29
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.
12
votes
3answers
3k views
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
For all $a, m, n \in \mathbb{Z}^+$,
$$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
31
votes
8answers
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 ...
14
votes
7answers
916 views
Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
Why is
$$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$
Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is
$$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$
This is ...
2
votes
2answers
630 views
Countable Sets and the Cartesian Product of them
If I have two countable sets, $A$ and $B$, how can I prove that the cartesian product of them, $A \times B$, is also countable?
17
votes
3answers
1k views
The set of differences for a set of positive Lebesgue measure
Quite a while ago, I heard about a statement in measure theory, that goes as follows:
Let $A \subset \mathbb R^n$ be a Lebesgue-measurable set of positive measure. Then we follow that $A-A = \{ ...
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 ...
15
votes
11answers
2k views
$ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
2
votes
1answer
278 views
Solving a recurrence relation with the characteristic equation
I have some trouble solving this due to not seeing the steps to be able to feed it into the characteristic equation.
$$T(n) = 4T(n-2) +n + 2^nn^2\ \text{with}\ \ T(0)=0,\ T(1)=1$$
(don't have to ...
2
votes
2answers
268 views
Units and Nilpotents
If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit.
I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) ...
13
votes
4answers
711 views
Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$
Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$
So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called ...
22
votes
3answers
10k 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 ...
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 $A = 2, C = 3$.
$$\frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} ...
1
vote
4answers
438 views
How to use fundamental theorem of arithmetic to conclude that $\gcd(a^k,b^n)=1$ for all $k, n \in$ N whenever $a,b \in$ N with $\gcd(a,b)=1$?
How to use fundamental theorem of arithmetic to conclude that $\gcd(a^k,b^n)=1$ for all $k, n \in$ N whenever $a,b \in$ N with $\gcd(a,b)=1$?
Fundamental theorem of arithmetic: Each number $n\geq 2$ ...
14
votes
9answers
4k views
How to show $\det(AB) =\det(A)\det(B)$
Given two square matrices $A$ and $B$. How do you show $\det(AB) = \det(A)\det(B)$ where $\det(\cdot)$ is determinant of the matrix
11
votes
4answers
757 views
Proof that there are infinitely many prime numbers starting with a given digit string
To prove the following fact: given any sequence of digits in any base, eg 314159265358979323 base 10, there are infinitely many primes that start with these digits,eg when expressed in decimal they ...
3
votes
1answer
104 views
How to prove a sequence of a function converges uniformly ???
For n $\in \mathcal{N}$, define the formula, $$f_n(x)= x/(2n^2x^2+8), x \in [0,1].$$ Prove that the sequence $f_n$ converges uniformly on [0,1], as n $\rightarrow \infty$.
I know that the definition ...
2
votes
1answer
286 views
Prove analyticity by Morera's theorem
Let $f$ be continuous on the complex plane and analytic on the complement of the coordinate axes. Show that $f$ is analytic everywhere. Hint: Morera's theorem. I think that I need to show that the ...
5
votes
1answer
452 views
Inequality involving $\limsup$ and $\liminf$
This may have been asked before, however I was unable to find any duplicate.
This comes from pg. 52 of "Mathematical Analysis: An Introduction" by Browder. Problem 14:
If $(a_n)$ is a sequence in ...
39
votes
4answers
1k views
Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even
I have a question:
Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then prove that
$$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$
How can I ...
11
votes
2answers
1k views
Automorphism of the Field of rational functions
Let $K$ be a field and let $K(x)$ be the field of rational functions in $x$ whose coefficients are in $K$. Let $\theta(x)$ $\in$ Aut($K(x))$ such that $\theta|_K = id_K$. Show that $\theta(x) ...
19
votes
3answers
504 views
If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$
I'm trying to prove the following:
If $(a_n)$ is a sequence of positive numbers such that $\sum_{n=1}^\infty a_n b_n<\infty$ for all sequences of positive numbers $(b_n)$ such that ...
7
votes
4answers
572 views
How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$?
I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it ...
4
votes
3answers
192 views
Indefinite integral of secant cubed
I need to calculate the following indefinite integral:
$$I=\int \frac{1}{\cos^3(x)}dx$$
I know what the result is (from Mathematica):
$$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$
but I don't ...
2
votes
2answers
140 views
prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
Let $f$ be a real-valued function continuous on $[a,b]$ and differentiable on $(a,b)$.
Suppose that $\lim_{x\rightarrow a}f'(x)$ exists.
Then, prove that $f$ is differentiable at $a$ and ...
2
votes
1answer
439 views
Recurrence relation, Fibonacci numbers
$(a)$ Consider the recurrence relation $a_{n+2}a_n = a^2
_{n+1} + 2$ with $a_1 = a_2 = 1$.
$(i)$ Assume that all $a_n$ are integers. Prove that they are all odd and the
integers $a_n$ and $a_{n+1}$ ...
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 ...
7
votes
1answer
268 views
Starting digits of 2^n
Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits.
5
votes
5answers
520 views
Largest integer that can't be represented as a non-negative linear combination of $m, n = mn - m - n$? Why?
This seemingly simple question has really stumped me:
How do I prove that the largest integer that can't be represented with a non-negative linear combination of the integers $m, n$ is $mn - m - n$, ...
4
votes
1answer
253 views
Operator whose spectrum is given compact set
Let $A\subset \mathbb{C}$ be a compact subset.
Since $A$ is compact and metric space, it is separable, say $\overline{\lbrace a_n\rbrace_{n=1}^\infty}=A$.
Let $\mathcal{l}^2(\mathbb{Z})$ be the ...
3
votes
3answers
479 views
A number when successively divided by $9$, $11$ and $13$ leaves remainders $8$, $9$ and $8$ respectively
A number when successively divided by $9$, $11$ and $13$ leaves remainders $8$, $9$ and
$8$ respectively.
The answer is $881$, but how? Any clue about how this is solved?
2
votes
3answers
329 views
How do you solve the Initial value probelm $dp/dt = 10p(1-p), p(0)=0.1$?
The problem is...
$$ \frac{dp}{dt} = 10p(1-p),$$ $p(0)=0.1$.
Solve and show that $p(t) \to 1$ as $t\to \infty.$
I know this is probably really simple, I was trying to go down the line of finding ...
1
vote
6answers
487 views
Integral of $\int e^{2x} \sin 3x\, dx$
I am suppose to use integration by parts but I have no idea what to do for this problem
$$\int e^{2x} \sin3x dx$$
$u = \sin3x dx$ $du = 3\cos3x$
$dv = e^{2x} $ $ v = \frac{ e^{2x}}{2}$
From this I ...
-2
votes
3answers
622 views
How to show that $\gcd(ab,n)=1$?
Let $\gcd(a,n)=\gcd(b,n)=1$. How to show that $\gcd(ab,n)=1$? This is a problem that is an exercise in my course.
16
votes
2answers
1k views
The union of a strictly increasing sequence of $\sigma$-algebras is not a $\sigma$-algebra
The union of a sequence of $\sigma$-algebras need not be a $\sigma$-algebra, but how do I prove the stronger statement below?
Let $\mathcal{F}_n$ be a sequence of
$\sigma$-algebras. If the ...
10
votes
4answers
786 views
Help with summing a power series
I'd like to determine the function corresponding to the following power series:
$$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1},
$$
where ...
12
votes
4answers
1k views
Computing the integral of $\log (\sin x)$
How to compute the following integral
$$\int \log(\sin x)\,\mathrm dx?$$
8
votes
2answers
782 views
Finding the adjoint of an operator
This is from my homework, I'm totally lost as to how to proceed.
Consider the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined by
$(Tf)(x) = \int^x_0 f(s) \ ds$
What is the adjoint of $T$?
...
7
votes
2answers
137 views
Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.
What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve ...
3
votes
2answers
764 views
Subadditivity of the limit superior
$$ \limsup \left(f(h)+g(h)\right) \leq \limsup f(h)+ \limsup g(h).$$
How can we prove this? Any help would be appreciated.
9
votes
6answers
997 views
Show that the set of all finite subsets of $\mathbb{N}$ is countable.
Show that the set of all finite subsets of $\mathbb{N}$ is countable.
I'm not sure how to do this problem. I keep trying to think of an explicit formula for 1-1 correspondence like adding all the ...
3
votes
5answers
474 views
Determine the Set of a Sum of Numbers
I want to determine the set of natural numbers that can be expressed as the sum of some non-negative number of 3s and 5s.
$$S=\{3k+5j∣k,j∈\mathbb{N}∪\{0\}\}$$
I want to check whether that would be:
...
2
votes
1answer
71 views
prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.
A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ R_{n}$ and $V ⊂ R_{m}$ are homeomorphic, then n = m. prove that the sphere with a ...
8
votes
4answers
1k views
Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]
Possible Duplicate:
Units and Nilpotents
Hi all wondering if I could get a bit of help with this, given $A^{2012}=0$ prove $(A+I)$ is invertible and find an expression for $(A+I)^{-1}$ in ...
6
votes
4answers
885 views
Equivalent metrics determine the same topology
Suppose that there are given two distance functions $d(x,y)$ and $d_1 (x,y)$ on the same space $S$. They are said to be equivalent if they determine the same open sets.
Show that $d$ and $d_1$ are ...
5
votes
3answers
414 views
Combinatorial proof for two identities
Does exist a combinatorial proof for the following two identities ?
$\sum_{k = 0}^{n} \binom{x+k}{k} = \binom{x+n+1}{n}$
$\sum_{k = 0}^{n} k\binom{n}{k} = n2^{n-1}$
I know how to derive the ...
1
vote
4answers
485 views
How can I prove that all rational numbers are either terminally real or repeating real numbers?
I am trying to figure out how to prove that all rational numbers are either terminally real or repeating real numbers, but I am having a great difficulty in doing so.
Any help will be greatly ...
5
votes
4answers
378 views
Finding $\int e^{2x} \sin{4x} \, dx$
Finding $$\int e^{2x} \sin 4x \, dx$$
I think I should be doing integration by parts...
If I let $u=e^{2x} \Rightarrow du = 2e^{2x}$,
$dv = \sin{4x} \Rightarrow v = -\frac{1}{4} \cos{4x}$
$\int{ ...
