Intereting Posts

How to prove that $e = \lim_{n \to \infty} (\sqrt{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt{n\#} $?
Is my understanding of quotient rings correct?
Solving this SDE $dX_t = aX_tdt + bdW_t$, $X_0 = x$ to find $E$
Sequences of sums of Pascal's triangle
Proving that $f(x,y) = \frac{x^2y}{x^2+y^2}$ with $ f(0,0)=0$ is continuous
Software to draw links or knots
Mean Value property for harmonic functions on regions other than balls/spheres
Intersection of open affines can be covered by open sets distinguished in *both*affines
Probability without replacement question
Is “$a + 0i$” in every way equal to just “$a$”?
What's the difference between $\mathbb{R}^2$ and the complex plane?
Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$
Prove that the composition of differentiable functions is differentiable.
A problem on range of a trigonometric function: what is the range of $\frac{\sqrt{3}\sin x}{2+\cos x}$?
$2^{4n+1} \equiv 1 \pmod{8n+7}$, this has been bugging me

**Question** : Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$.

**Hint:**

My solution doesn’t use continued fraction.

- Preparing for Spivak
- Use $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$ to compute $\sum_{n=1}^\infty \frac{(-1)^n}{n^4}$
- Differentiation under integral sign (Gamma function)
- limit of a recursively defined function
- Given $f\notin L^p$ find $g\in L^q$ s.t. $fg\notin L^1$
- What's the relationship between a measure space and a metric space?

I am interested in other possible proofs for this question.

- Evaluation of the integral $\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$
- For a continuous function $f$ and a convergent sequence $x_n$, lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$
- Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt{c_n}\leq\lim\sup\sqrt{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$
- Differentiation under the integral sign for Lebesgue integrable derivative
- When is the difference of two convex functions convex?
- Cluster points of multiples of the fractional part of an irrational number.
- How to make a smart guess for this ODE
- Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$
- Integral eigenvectors and eigenvalues
- Showing that the norm of the canonical projection $X\to X/M$ is $1$

Suppose that $\arctan n=r\in\Bbb Q$, where $n$ is a non-zero integer. Then $r$ is not zero, so $2r$ is not zero, and

$$\cos2r=\frac{\cos^2r-\sin^2r}{\cos^2r+\sin^2r}

=\frac{1-\tan^2r}{1+\tan^2r}=\frac{1-n^2}{1+n^2}$$

which is rational. But this contradicts the result that the cosine of a non-zero rational number is irrational.

As for the proof of this result, it is usually done by taking an integral such as

$$\int_0^r f(x)\sin x\,dx\ ,$$

where $f(x)=x^n(a-bx)^{2n}(2a-bx)^n$ and $r=a/b$, and showing that if $n$ is large we get contradictory estimates for the integral. See, for example, my lecture notes, starting at page 20.

- List of explicit enumerations of rational numbers
- Prove: If a sequence converges, then every subsequence converges to the same limit.
- Using Fermat's little theorem to find remainders.
- Are derivatives defined at boundaries?
- How to show that the monomials are not a Schauder basis for $C$
- Areas versus volumes of revolution: why does the area require approximation by a cone?
- A special cofinal family in $(\omega^\omega,\le)$
- Geometric interpretation of hyperbolic functions
- On the order of elements of $GL(2,q)$?
- How can I evaluate this indefinite integral? $\int\frac{dx}{1+x^8}$
- a.s. Convergence and Convergence in Probability
- Find the sum $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + …+ \frac{1}{\sqrt{99}+\sqrt{100}}$
- How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $?
- convergence in probability induced by a metric
- Find all irreducible monic polynomials in $\mathbb{Z}/(2)$ with degree equal or less than 5