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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52
votes
13answers
5k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
54
votes
12answers
6k 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 ...
27
votes
4answers
5k views

Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$

For all $a, m, n \in \mathbb{Z}^+$, $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
20
votes
9answers
3k 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 ...
33
votes
5answers
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.
4
votes
3answers
3k 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
799 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 ...
20
votes
12answers
3k 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?
13
votes
6answers
4k 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 ...
5
votes
1answer
746 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 ...
3
votes
2answers
768 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) ...
4
votes
2answers
1k views

Cartesian Product of Two Countable Sets is Countable

How can I prove that the Cartesian product of two countable sets is also countable?
15
votes
6answers
3k views

Computing the integral of $\log(\sin x)$

How to compute the following integral $$\int\log(\sin x)~dx~?$$
35
votes
3answers
12k 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 ...
5
votes
3answers
1k views

Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
22
votes
3answers
2k 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 = \{ ...
11
votes
5answers
2k 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 ...
22
votes
3answers
750 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 ...
8
votes
2answers
689 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 ...
1
vote
4answers
701 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 $n\geq 2$ ...
11
votes
4answers
595 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 ...
5
votes
7answers
501 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 ...
5
votes
5answers
732 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: ...
8
votes
5answers
1k 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} ...
2
votes
4answers
2k 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 ...
9
votes
4answers
2k views

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
3
votes
1answer
166 views

Determine if the following series are convergent or divergent?

How to determine if the following series are convergent or divergent? I'm supposed to use here the limit comparison test, but I don't know how to choose the second series. $$\sum_{k=1}^\infty \ln(1+ ...
21
votes
4answers
2k 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 ...
12
votes
5answers
5k views

Limit of the derivative of a function as x goes to infinity

I've been trying to solve the following problem: Suppose that $f$ and $f'$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and ...
9
votes
4answers
2k views

How does the existence of a limit imply that a function is uniformly continuous

I am working on a homework problem from Avner Friedman's Advanced Calculus (#1 page 68) which asks Suppose that $f(x)$ is a continuous function on the interval $[0,\infty)$. Prove that if ...
7
votes
2answers
466 views

The Center of $\operatorname{GL}(n,k)$

The given question: Let $k$ be a field and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$. I have spent a while trying to prove this ...
8
votes
1answer
323 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.
3
votes
3answers
629 views

Irreducibility of $X^{p-1} + \ldots + X+1$

Can someone give me a hint how to the irreducibility of $X^{p-1} + \ldots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ? Our professor gave us already one, namely to substitute $X$ with $x+1$, but ...
45
votes
4answers
2k 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 ...
21
votes
10answers
12k 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
18
votes
2answers
2k 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 ...
7
votes
6answers
723 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 ...
6
votes
4answers
201 views

$M,N\in \Bbb R ^{n\times n}$, show that $e^{(M+N)} = e^{M}e^N$ given $MN=NM$

I am working on the following problem. Let $e^{Mt} = \sum\limits_{k=0}^{\infty} \frac{M^k t^k}{k!}$ where $M$ is an $n\times n$ matrix. Now prove that $$e^{(M+N)} = e^{M}e^N$$ given that $MN=NM$, ie ...
2
votes
1answer
2k views

Subgroup of index 2 is Normal

Please excuse the selfishness of the following question: Let $G$ be a group and $H \le G$ such that $|G:H|=2$. Show that $H$ is normal. Proof: Because $|G:H|=2$, $G = H \cup aH$ for some $a \in ...
3
votes
3answers
1k 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?
1
vote
6answers
1k 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 ...
19
votes
2answers
3k views

Why is the ring of matrices over a field simple?

Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$. Then why does this ring not contain any two-sided ideal? Thanks for any clarification, and this is ...
10
votes
3answers
2k views

A ring that is not an Euclidean domain

Let $\alpha = \frac{1+\sqrt{-19}}{2}$. Let $A = \mathbb Z[\alpha]$. Let's assume that we know that its invertibles are $\{1,-1\}$. During an exercise we proved that: Lemma: If $(D,g)$ is an ...
10
votes
4answers
3k views

Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
1
vote
3answers
213 views

How do I compute $\int_{-\infty}^\infty e^{-\frac{x^2}{2t}} e^{-ikx} \, \mathrm dx$ for $t \in \mathbb{R}_{>0}$ and $k \in \mathbb{R}$?

Let $t \in \mathbb{R}_{>0}$ and $k \in \mathbb{R}$. I want to find $$\int_{-\infty}^\infty e^{-\frac{x^2}{2t}} e^{-ikx} \, \mathrm dx.$$ A hint told me to first determine $\int_{-\infty}^\infty ...
0
votes
1answer
174 views

For what values $\alpha$ for complex z $\ln(z^{\alpha}) = \alpha \ln(z)$?

For example, when $\alpha = 2$, $\ln(z^{2}) \neq 2\ln(z)$, because argument z is determined up to constant $2 \pi k$. So $$ \ln(z^{2}) = \ln(z) + \ln(z) = \ln(z_{k_{1}}) + \ln(z_{k_{2}}) \neq ...
5
votes
1answer
356 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 ...
2
votes
1answer
130 views

Prove or find a counter example to the claim that for all sets A,B,C if A ∩ B = B ∩ C = A ∩ C = Ø then A∩B∩C ≠ Ø

This is a homework question of mine that Ive answered and Id really like some feedback on my solution. Prove or find a counter example to the claim that for all sets $A,B,C$, if $A\cap B=B\cap ...
2
votes
5answers
324 views

Finding generating function for the recurrence $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$

I am trying to find generating function for the recurrence: $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$ for every $n \ge 1$. It looks like this: $a_0 = 1$ $a_1 = {1 \choose 2} + 3$ $a_2 = {2 ...
-5
votes
3answers
135 views

Give an example of a function $f: \mathbb{N} \rightarrow \mathbb{N}$ with the property that there exists [closed]

Give an example of a function $f: \mathbb{N} \rightarrow \mathbb{N}$ with the property that there exists a function $g: \mathbb{N} \rightarrow \mathbb{N}$ such that the composition $g \circ f$ is the ...