marty cohen
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Show that this sum is an integer.
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46 votes

Want $\sum_{k=1}^{n-1} g\left(\frac{k}{n}\right) $ where $n$ is odd and $g(t)=\dfrac{3^t}{3^t+3^{1/2}} $. $g(k/n) =\dfrac{3^{k/n}}{3^{k/n}+3^{1/2}} $. $\begin{array}\\ g(k/n)+g((n-k)/n) &=\dfrac{...

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Almost Stirling's Approximation
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37 votes

So you have $n! \lt \left( \frac{n+1}{e} \right)^n \sqrt{n+1} $. I'll play and see what happens. Replacing $n$ by $n-1$, we get $(n-1)! \lt \left( \frac{n}{e} \right)^{n-1} \sqrt{n} $. ...

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Is the binomial theorem actually more efficient than just distributing
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36 votes

The binomial theorem allows you to write out the expansion of your polynomial immediately. It also allows you to answer such questions as "What is the coefficient of $x^{20}$ in $(1+x)^{100}$?" Its ...

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Is there a website like OEIS for real constants?
Accepted answer
34 votes

Inverse symbolic calculator plus: https://isc.carma.newcastle.edu.au/

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Mistaken counterexample to FLT; where's the mistake?
Accepted answer
26 votes

$(10a+1)^n$ always ends in $1$ and $(10b+6)^n$ always ends in $6$ so $(10a+1)^n+(10b+6)^n$ always ends in $7$.

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Why doesn't integrating the area of the square give the volume of the cube?
26 votes

Actually, you have two errors there: The minor one is that you seem to want a cube of side $2r$, since your integral goes from $-r$ to $r$. The major error, as others have said, is that you are ...

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Proof of $\lim_{n\to \infty} \sqrt[n]{n}=1$
26 votes

Here's a two-line, completely elementary proof that uses only Bernoulli's inequality: $$(1+n^{-1/2})^n \ge 1+n^{1/2} > n^{1/2}$$ so, raising to the $2/n$ power, $$ n^{1/n} < (1+n^{-1/2})^2 = 1 ...

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How to prove that the set of rational numbers are countable?
25 votes

Map each rational $\frac{a}{b}$ into the integer $2^a 3^b$. This shows that the number of rationals is at most the number of integers. If you want to handle the negative rationals, map the sign ($-1$,...

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Find the maximum of $f(x)=x^{1/x}$
24 votes

This is not original with me. If we know that $e^x \ge 1+x$ with equality only when $x = 0$, $e^{(x-e)/e} \ge 1 + (x-e)/e = x/e$ or $e^{x/e} \ge x$ or $e^{1/e} \ge x^{1/x}$ with equality only if $x = ...

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How is the derivative truly, literally the "best linear approximation" near a point?
21 votes

I'll first give a intuitive answer, then an analytic answer. Intuitively, the tangent goes in the same direction as the function, following it as closely as possible for a line. Any other line ...

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Dividing an infinite power series by another infinite power series
20 votes

The standard way (in other words, there is nothing original in what I am doing here) to get $H(x)$ is to write $H(x)G(x) = F(x)$ and get an iteration for the $c_n$. $\begin{array}\\ H(x)G(x) &=\...

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pronunciation of sinh x, cosh x, tanh x for short
20 votes

I usually say "sine-h", "cos-h", and "tan-h" with the "h" pronounced "aich" like the letter. Sometimes I pronounce "cosh" as a word with a long "o". I guess this qualifies as an answer, instead of ...

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Perfect powers of successive naturals: Can you always reach a constant difference?
18 votes

Yes. The first difference of a polynomial of degree $d$ is a polynomial of degree $d-1$. By induction, the $m$-th difference of a polynomial of degree $d$, when $m \le d$, is a polynomial of degree $...

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What could be better than base 10?
17 votes

I like the factorial base, where the integer part of a real number is written as $\sum_{i=2}^n a_i i!$ where the $a_i$ are integers such that $0 \le a_i < i$ and the fractional part is written as $\...

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Why don't we define "imaginary" numbers for every "impossibility"?
16 votes

The reason why the complex numbers are so special is that they are the end of a chain of questions of the form "How can we solve this equation?" or "What are the roots of this equation?". We start ...

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Theorems with an extraordinary exception or a small number of sporadic exceptions
16 votes

Vinogradov's theorem that all but a finite number of odd numbers are the sum of three primes. Not sure about the current state of Goldbach's conjecture. A theorem at about my level of math: All ...

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Visual representation of the fact that there are more irrational than rational numbers.
15 votes

The rationals can be mapped into the lattice points $(n, m)$, which are an infinite set of isolated points in the plane. The irrationals, by any of the standard ways of mapping two reals into one (...

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Understanding “the mean minimizes the mean squared error”
14 votes

If you have $(y_i)_{i=1}^n$, consider the mean squared difference from the $y_i$ to a value $a$. This is $s(a) =\sum_{i=1}^n (y_i-a)^2 $. Manipulating this, $\begin{array}\\ s(a) &=\sum_{i=1}^...

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What are the possible values of these letters?
14 votes

A start, on my phone. Assume $j<k<l<m<n.$ Then j=2 or 3 because 1 makes the sum too large and 4 makes it too small. Therefore the left without 1/j is 1/2 or 2/3. You can get a tree of ...

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Proof of the infinite descent principle
Accepted answer
13 votes

I would argue a different way. By assumption, for all $n$, $a_n > a_{n+1}$, or $a_n \ge a_{n+1}+1$. Therefore, since $a_{n+1} \ge a_{n+2}+1$, $a_n \ge a_{n+2}+2$. Proceeding by induction, for ...

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None of $3,5,7$ can divide $r^4+1$
13 votes

I'm going to try to make the most elementary possible proof. $\bmod 3$, $(0, \pm 1)^4 \equiv (0, 1) $ so $(0, \pm 1)^4+1 \equiv (1, 2) $ so $3 \not \mid r^4+1$. $\bmod 5$, $(0, \pm 1, \pm 2)^4 \...

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Aren't there obvious patterns in the primes that no one makes use of and what about this...
Accepted answer
13 votes

This is an example of a sieve. It is well known, and I used it over forty years in a program to (iirc) check for primality. Yes, the pattern continues. No, it cannot be used to prove the twin prime ...

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Conway's "Murder Weapon"
12 votes

A Google search for "Conway and Coxeter murder" came up with the book "King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry". A search via Google inside this book came up with this on ...

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Can a sequence have infinitely many limits among its subsequences?
12 votes

$\sin(n) _{n=1}^{\infty}$ (originally entered $\sin(1/n)$, because I was thinking of $\sin(1/x)$ as $x \to 0$.)

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Why does $\lim_{n \to \infty} \sqrt{\frac{n}{n+1}} = 1$?
12 votes

It is very well determined. $\lim_{n \to \infty} \frac{n}{n+1} = 1$ (since $1-\frac{n}{n+1} = \frac{1}{n+1} \to 0$). Since $\sqrt{}$ is continuous at $1$, if $f(n) \to 1$ as $n \to \infty$, $\sqrt{f(...

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What is the correct spelling of Paul Erdős's name?
12 votes

As for the pronunciation, just remember this (not by me, but I do not know who the author is): $\begin{align} &\text{A theorem both deep and profound}\\ &\text{states that "Every circle is ...

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What is lost when we move from reals to complex numbers?
11 votes

Comparing naturals to integers, there is a smallest natural (0 or 1) which often makes solving problems easier. Comparing reals to complex, you can always compare reals but there is no complete ...

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Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.
11 votes

Many questions with sum or difference of square roots can be solved with conjugating. So, if $s = \sqrt{4 + 2\sqrt{3}} - \sqrt{3}$, and $t = \sqrt{4 + 2\sqrt{3}} + \sqrt{3}$, $\begin{array}\\ st &...

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Formula for $1! \times 2! \times \cdots \times n!$?
11 votes

In the May 2013 Fibonacci Quarterly (Vol. 51, Num. 2) pages 163-173, Michael Hirschhorn proved this result: Let $P(n) =\prod\limits_{k=0}^n \binom{n}{k} $. Then (this is going to be a pain to enter), ...

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Limit of $\prod\limits_{k=2}^n\frac{k^3-1}{k^3+1}$
Accepted answer
11 votes

Using the suggested factorizations, and using $\begin{array}\\ k^2-k+1 &=k(k-1)+1\\ &=(k-1+1)(k-1)+1\\ &=(k-1)^2+(k-1)+1\\ \end{array} $ (this is really the key), $\begin{array}\\ \...

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