80
votes
20answers
5k views

Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

I recently proved that $$ \sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2 $$ Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
19
votes
5answers
4k 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 ...
50
votes
5answers
2k views

Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
19
votes
4answers
2k views

Sine function dense in $[-1,1]$

We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by ...
19
votes
6answers
2k views

How to solve $x^3=-1$?

How to solve $x^3=-1$? I got following: $x^3=-1$ $x=(-1)^{\frac{1}{3}}$ $x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
86
votes
7answers
8k views

Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
41
votes
2answers
2k views

Identity for convolution of central binomial coefficients

It's not difficult to show that $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$ On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing ...
29
votes
2answers
8k views

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, ...
14
votes
4answers
1k 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, ...
3
votes
3answers
169 views

Prove $\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$

Could someone explain to me why the identity $$ \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k} $$ holds?
132
votes
18answers
13k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
42
votes
9answers
6k views

Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$

I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$ What do you think? It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm ...
30
votes
2answers
20k views

Expected time to roll all 1 through 6 on a die

What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die? The order in which the numbers appear does not matter. I had this questions explained ...
32
votes
5answers
3k views

Dominoes and induction, or how does induction work?

I've never really understood why math induction is supposed to work. You have these 3 steps: Prove true for base case (n=0 or 1 or whatever) Assume true for n=k. Call this the induction ...
4
votes
1answer
1k 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 ...
60
votes
18answers
3k views

Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very ...
31
votes
2answers
2k views

Set of continuity points of a real function

I have a question about subsets $$ A \subseteq \mathbb R $$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize ...
37
votes
2answers
1k views

Evaluating $\int P(\sin x, \cos x) \text{d}x$

Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$. For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$. Is there a general method which allows us to evaluate the ...
16
votes
8answers
997 views

How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$?

I've spent the better part of this day trying to show from first principles that this sequence tends to 1. Could anyone give me an idea of how I can approach this problem? $$ \lim_{n \to +\infty} ...
14
votes
5answers
671 views

Given $y_n=(1+\frac{1}{n})^{n+1}$ show that $\lbrace y_n \rbrace$ is a decreasing sequence

Given $$ y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1. $$ Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ? I consider the ...
5
votes
1answer
598 views

A few improper integral

$$\displaystyle \begin{align*} & \int_{0}^{+\infty }{\frac{\text{d}x}{1+{{x}^{n}}}} \\ & \int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x} \\ & \int_{0}^{+\infty ...
8
votes
2answers
1k 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) ...
36
votes
4answers
5k views

Construction of a Borel set with positive but not full measure in each interval

I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere. To be precise, if $\mu$ denotes Lebesgue ...
47
votes
16answers
14k views

Why is negative times negative = positive?

Someone recently asked me why a negative * a negative is positive, and why a negative * a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Assume $(-x) * (-y) ...
19
votes
15answers
3k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
14
votes
2answers
2k views

Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means

Prove that if $\lim_{n \to \infty}z_{n}=A$ then: $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: ...
3
votes
1answer
1k views

$\sum \cos$ when angles are in arithmetic progression [duplicate]

Possible Duplicate: How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + ...
194
votes
12answers
56k 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? ...
44
votes
2answers
2k views

Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$

Consider the sequence defined as $x_1 = 1$ $x_{n+1} = \sin x_n$ I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't ...
39
votes
10answers
4k views

The Monty Hall problem

I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the ...
22
votes
6answers
1k views

Simpler way to compute a definite integral without resorting to partial fractions?

I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$ Is there a simpler way to do this ?
20
votes
1answer
2k 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 ...
9
votes
4answers
3k views

Fibonacci modular results

Can any one give a generalization of the following properties in a single proof? I have checked the results, which I have given below by trial and error method. I am looking for a general proof, which ...
353
votes
10answers
369k views

Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
88
votes
18answers
15k views

Striking applications of integration by parts

What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your ...
47
votes
9answers
1k views

Does $G\cong G/H$ imply that $H$ is trivial?

Let $G$ be any group such that $$G\cong G/H$$ where $H$ is a normal subgroup of $G$. If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In ...
58
votes
9answers
6k views

Proof that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$

I was wondering the following. And I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function ...
32
votes
7answers
2k views

Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
64
votes
10answers
3k views

Why 1 is not considered to be a prime number?

Why $1$ is not considered to be a prime number? Or why definition of prime numbers is given for integers greater than $1$?
50
votes
4answers
2k views

“Closed” form for $\sum \frac{1}{n^n}$

Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating ...
12
votes
7answers
7k views

The product of n consecutive integers is divisible by n factorial

How can we prove that the product of n consecutive integers is divisible by n factorial? Note: In this subsequent question and the comments here the OP has clarified that he seeks a proof that ...
9
votes
3answers
6k views

How to prove a sequence of a function converges uniformly?

For $n \in \mathbb{N}$, define the formula, $$f_n(x)= x/(2n^2x^2+8),\quad x \in [0,1].$$ Prove that the sequence $f_n$ converges uniformly on $[0,1]$, as $n \to \infty$. I know that the definition ...
7
votes
6answers
2k views

Comaximal ideals in a commutative ring

Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r ...
38
votes
10answers
20k views

How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?

Could you provide a proof of Euler's formula: $e^{it}=\cos t +i\sin t$ ? thanks.
46
votes
17answers
4k views

In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?

I am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is: ...
21
votes
3answers
3k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
36
votes
8answers
7k views

Taking Seats on a Plane

This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless Imagine there are a 100 people in line to ...
26
votes
2answers
1k views

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

Is there any way to show that $$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
28
votes
8answers
1k views

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$

why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$? I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward. ...
26
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?

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