Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [faq]

This is meant for questions which are generalized forms of questions which get asked frequently. See tag details for more information.

26
votes
7answers
3k views

How can a function with a hole (removable discontinuity) equal a function with no hole?

I've done some research, and I'm hoping someone can check me. My question was this: Assume I have the function $f(x) = \frac{(x-3)(x+2)}{(x-3)}$, so it has removable discontinuity at $x = 3$. We ...
32
votes
7answers
5k views

Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero? [duplicate]

If $a_n$ is a sequence such that $$a_1 \leq a_2 \leq a_3 \leq \dotsb$$ and has the property that $a_{n+1}-a_n \to 0$, then can we conclude that $a_n$ is convergent? I know that without the ...
0
votes
1answer
44 views

Typing mathematical notation [closed]

Somebody please provide me the easiest and fastest way to type mathematical notation, I am new here please help.
4
votes
2answers
6k views

How to tell whether two graphs are isomorphic?

Suppose that we are given two graphs on a relatively small number of vertices. Here's an example: Are these two graphs isomorphic? (That is: is there a bijection $f$ from $\{A,B,\dots,I\}$, the ...
3
votes
1answer
2k views

Eigenvalues of the principal submatrix of a Hermitian matrix

This question aims at creating an "abstract duplicate" of various questions that can be reduced to the following: Let $A$ be an $n\times n$ Hermitian matrix and $B$ be an $r\times r$ principal ...
4
votes
4answers
511 views

Find $6^{1000} \mod 23$ [duplicate]

Find $6^{1000} \mod 23 $ Having just studied Fermat's theorem I've applied $6^{22}\equiv 1 \mod 23 $, but now I am quite clueless on the best way to proceed. This is what I've tried: Raising ...
4
votes
1answer
488 views

How Do I Compute the Eigenvalues of a Small Matrix?

If I have a $2\times 2$ or $3\times 3$ matrix, how should I go about computing the eigenvalues and eigenvectors of the matrix? NB: I am making this question to provide a unified answer to questions ...
0
votes
1answer
88 views

Sterling Numbers of The Second Kind With Limitations Placed on Boxes/Parts

I know there are similar problems already on the board. However, none of the previously stated questions contain problems where limitations are placed on the BOXES. Thus, seeing that I am struggling ...
45
votes
13answers
8k views

Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$

After reading this question, the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$ What's the name of this identity? Is it the identity of the Pascal's triangle ...
18
votes
8answers
2k views

Summation Theorem how to get formula for exponent greater than 3

I'm studying in the summer for calculus 2 in the fall and I'm reading about summation. I'm given these formulas: \begin{align*} \sum_{i=1}^n 1 &= n, \\ \sum_{i=1}^n i &= \frac{n(n+1)}{2},\\ \...
12
votes
1answer
1k views

Volume of the intersection of two cylinders

I have two infinite cylinders of unit radius in $\mathbb{R}^3$, whose axes are skew lines. Say that the axis of one is centered on the $x$-axis, and the axis of the other is determined by the two ...
1
vote
2answers
5k views

Find $11^{644} \mod 645$ [duplicate]

Can someone just explain to me the basic process of what is going on here? I understand everything until we start adding 1's then after that it all goes to hell. I just need some guidance. The Problem ...
0
votes
1answer
1k views

Volume of Intersection of cylinders (different radii)

I want to derive a formula for the area of the intersection of two rigth cylinders with different radii. To get an idea I attached a sketch. My idea is to determine the borders of $x$ and $z$ in ...
3
votes
4answers
1k views

How to compute $3^{2003}\pmod {99}$ by hand? [duplicate]

Compute $3^{2003}\pmod {99}$ by hand? It can be computed easily by evaluating $3^{2003}$, but it sounds stupid. Is there a way to compute it by hand?
3
votes
2answers
363 views

How To Prove:$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4} = -\frac{7}{{720}}{\pi ^4}$

When I tried to solve this integral: $$\int_0^\infty {\frac{{{x^3}}}{{1 + {e^x}}}} \;{\rm{d}}x$$ I had trouble computing the sieries: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}$$ Thanks.
2
votes
3answers
381 views

Calculating volume enclosed using triple integral

Calculate the volume enclosed between $x^2 + y^2=1$, $y^2+z^2=1$, $x^2+z^2=1$ I am supposed to solve this question using multiple integral. I am not able to visualize the resultant figure of whose I ...
36
votes
1answer
3k views

Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Eigenvalues of a matrix of $1$'s ...
9
votes
1answer
4k views

Single Variable Calculus Reference Recommendations

This question is a generalization of the common question asking for calculus references. It is here to abstract away the repetition, and give a canonical resource for calculus references. I'm ...
3
votes
3answers
5k views

Find the Mean for Non-Negative Integer-Valued Random Variable

Let $X$ be a non-negative integer-valued random variable with finite mean. Show that $$E(X)=\sum^\infty_{n=0}P(X>n)$$ This is the hint from my lecturer. "Start with the definition $E(X)=\sum^\...
2
votes
5answers
2k views

Find a closed form for $\sum_{k=0}^{n} k^3$ [duplicate]

Find a closed form for $\sum_{k=0}^{n} k^3$. I would appreciate ideas for approaching questions like this in general as well. Thanks.
4
votes
1answer
2k views

Different ways of Arranging balls in boxes

This question is generalization of different cases of combinatorics problems that are generally asked. We will find general way of arranging $n$ balls in $r$ boxes. Cases : Identical Balls. ...
5
votes
3answers
346 views

For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$? [duplicate]

Let $ (G,\ast)$ be a group with identity $e$ and cardinality $2n$ for some $n\in\omega$. Then, does there exist $x\in G$ such that $x\ast x=e$ and $x\neq e$?
9
votes
3answers
33k views

How many equivalence relations on a set with 4 elements.

Let S be a set containing 4 elements (I choose {$a,b,c,d$}). How many possible equivalence relations are there? So I started by making a list of the possible relations: {$(a,a)(a,b)(a,c)(a,d)(b,a)(b,...
6
votes
7answers
2k views

Show that if $n>2$, then $(n!)^2>n^n$.

Show that if $n>2$, then $(n!)^2>n^n$. My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n>(n+1)^{n-1}$ Here, I tried to use induction again without any ...
4
votes
6answers
35k views

Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer)

Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer) I saw no example of this problem anywhere.. I saw an ...
29
votes
5answers
3k views

Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?

Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}...
68
votes
1answer
11k views

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...
11
votes
3answers
529 views

Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$

Let $a\ge 2$ be an integer. Show that for positive integers $m,n$, we have $a^n - 1$ divides $a^m - 1$ if and only if $n$ divides $m$. I am having trouble showing this. I've seen a similar problem on ...
4
votes
3answers
417 views

Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

I'm trying to prove rigorously that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$. Where $f$ is the pdf of the random variable $X$. I can't find a proof on the wikipedia article, or if it's ...
27
votes
9answers
13k views

About $\lim \left(1+\frac {x}{n}\right)^n$

I was wondering if it is possible to get a link to a rigorous proof that $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
24
votes
10answers
24k views

How to determine equation for $\sum_{k=1}^n k^3$

How do you find an algebraic formula for $\sum_{k=1}^n k^3$? I am able to find one for $\sum_{k=1}^n k^2$, but not $k^3$. Any hints would be appreciated.
150
votes
16answers
57k views

Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
25
votes
5answers
5k views

Can a finite sum of square roots be an integer?

Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer? The square roots need to be irrational.
32
votes
3answers
7k views

Multiples of an irrational number forming a dense subset

Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in $[0,1]...
12
votes
5answers
8k views

How to compute the determinant of a tridiagonal matrix with constant diagonals?

How to show that the determinant of the following $(n\times n)$ matrix $$\begin{pmatrix} 5 & 2 & 0 & 0 & 0 & \cdots & 0 \\ 2 & 5 & 2 & 0 & 0 & \cdots & ...
4
votes
4answers
473 views

Differentiation of $x^{\sqrt{x}}$, how?

The answer is (I think) $x^{\sqrt{x}-0.5} (1+0.5\ln(x))$, but how?
8
votes
1answer
1k views

Find volume of crossed cylinders without calculus.

I found this puzzle here. (It's labeled "crossed cylinders".) Here's the description: Two cylinders of equal radius are intersected at right angles as shown at left. Find the volume of the ...
33
votes
1answer
4k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
66
votes
7answers
35k 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 ...
10
votes
2answers
3k views

When are the limit operations commutative?

I'll come up with a question that has bothered me for a long period of time. The question seems relatively simple, but I personally didn't manage to find an answer to it. In many cases I met problems ...
30
votes
2answers
39k views

Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$

Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show, $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$ when $X$ has : a) a discrete distribution, b) a continuous ...
14
votes
2answers
4k views

Finite Sum of Power?

Can someone tell me how to get a closed form for $$\sum_{k=1}^n k^x$$ For $x = 1$, it's just the classic $\frac{n(n+1)}2$. What is it for $x > 1$?
52
votes
5answers
18k views

Highest power of a prime $p$ dividing $N!$

How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce abstract ...
16
votes
4answers
21k 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).
24
votes
4answers
5k 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) = 1 ...
65
votes
4answers
11k views

$\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$

(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12) For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x > 0$. Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and ...
58
votes
2answers
34k views

Continuous mapping on a compact metric space is uniformly continuous

I am struggling with this question: Prove or give a counterexample: If $f : X \to Y$ is a continuous mapping from a compact metric space $X$, then $f$ is uniformly continuous on $X$. Thanks for ...
4
votes
2answers
2k views

Converting multiplying fractions to sum of fractions

I have the next fraction: $$\frac{1}{x^3-1}.$$ I want to convert it to sum of fractions (meaning $1/(a+b)$). So I changed it to: $$\frac{1}{(x-1)(x^2+x+1)}.$$ but now I dont know the next step. ...
81
votes
9answers
10k views

How do I compute $a^b\,\bmod c$ by hand?

How do I efficiently compute $a^b\,\bmod c$: When $b$ is huge, for instance $5^{844325}\,\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, ...
58
votes
16answers
48k 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 being ...