# All Questions

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### 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 ...
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 ...
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### 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? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
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### 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 ...
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### How to solve $x^3=-1$?

How to solve $x^3=-1$? I got following: $x^3=-1$ $x=(-1)^{\frac{1}{3}}$ $x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
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### 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 $\frac 1p$ ...
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### 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 ...
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### 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 $L^q$, ...
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### 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, ...
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### 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?
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### 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 ...
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### 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 ...
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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? ...
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### 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 ...
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### 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 ...
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### 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 ?
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### 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 ...
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### 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 ...
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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ?
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### 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 $1$?
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### “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 ...
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### 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 ...
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### 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 ...
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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 $n_1,...,n_r ... 10answers 20k views ### How to prove Euler's formula:$e^{it}=\cos t +i\sin t$? Could you provide a proof of Euler's formula:$e^{it}=\cos t +i\sin t$? thanks. 17answers 4k views ### In classical logic, why is$(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$') is: ... 3answers 3k views ### Prove that$\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 $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$ 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 ... 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( ... 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. ... 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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