89 votes
Accepted

Does every Cauchy sequence converge to *something*, just possibly in a different space?

You are correct in the narrow sense that every Cauchy sequence does converge in some space. To be precise, let $(X;d_1)$ be any metric space with at least two points, let $Y$ be the set of Cauchy ...
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39 votes
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Historical Mistake of Assuming Measurability

There is the famous mistake of Lebesgue. Quoting from Descriptive Set Theory: Second Edition by Yiannis N. Moschovakis , page 2: "Lebesgue's argument was “simple, short but false.” The wrong ...
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  • 12.8k
20 votes
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Why $f(z+1)=f(z)$ implies $f$ can be expressed as a function of $e^{2\pi iz}$

Functions that are $1$-periodic like this can be considered as functions on $\Bbb{C} / \sim$, where $\sim$ is the equivalence relation $$z \sim w \iff z - w \in \Bbb{Z}.$$ If you like, $\Bbb{C}/ \sim$ ...
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  • 43.3k
19 votes

Estimating the value of $e$ using a random function

Generate $N$ random permutations (possibly with Knuth's shuffle). Then count how many of them are derangements: $N_D$. Then $$\frac N{N_D}\simeq e$$
19 votes
Accepted

$e^{-nx}\cdot\sum_{k=0}^\infty\frac{(nx)^k}{k!}f\left(\frac{k}{n}\right)\to f(x)$ for $f$ continuous and bounded

Here's a fun probabilistic proof. Let $X_1,X_2,\dots$ be iid Poisson random variables with parameter $x>0$. Then $S_n=X_1+\dots+X_n$ is Poisson with parameter $nx$. By the weak law of large numbers,...
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  • 8,492
16 votes

Doubt about the statement of the Fundamental Theorem of Algebra.

While the second version is logically stronger than the first version, it's not hard at all to get the second from the first: divide $p(z)$ by $z-z_0$, and it can be shown that the quotient is still a ...
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  • 7,450
16 votes
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Elementary proof of $\lim_{n\to\infty}n(\sqrt[n]{n}-1)=\infty$

For each $M \in \mathbb{R}$, choose positive integers $k$ and $N$ so that $\frac{k}{2} > M$ and $N = 2^k$. Then for each $n \geq N$, we have \begin{align*} n \bigl( n^{1/n} - 1 \bigr) &\geq n \...
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15 votes

$\int^{\infty}_{-\infty}u(x,y) \,d y$ independent of x

Although I'm about 7 years late, here is an answer anyway for anyone interested: Claim $$I = \frac{e}{2} \sqrt{\pi} \, \text{erfc} (1)$$ and is thus independent of $x$. Proof. By https://dlmf.nist....
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  • 3,064
14 votes

Better Proofs Than Rudin's For The Inverse And Implicit Function Theorems

I first studied the proof of the inverse function theorem over 20 years ago, and it seemed opaque. But now I finally understand the proof well enough that it seems clear and almost straightforward. I ...
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  • 48.2k
14 votes
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Evaluate $\int_{0}^{\infty} \ln(1+\frac{2\cos x}{x^2} +\frac{1}{x^4}) \, dx$

For $a>0$, let $F(a) = \int_{-\infty}^\infty \log(1+\frac{e^{aix}}{x^2}) dx$, your $I$ equals $\Re F(1)$. By parts gives $$F(a) = \int_{-\infty}^\infty \frac{i (a x+2 i)}{1+x^2 e^{-i a x}} dx $$ ...
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  • 17.8k
13 votes
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General form for $\sum_{n=1}^{\infty} (-1)^n \left(m n \, \text{arccoth} \, (m n) - 1\right)$

Let $$S\left( m \right)=\sum\limits_{n=1}^{\infty }{{{\left( -1 \right)}^{n}}\left( mn\operatorname{arcoth}\left( mn \right)-1 \right)},\,\,\left| m \right|>1$$ And note the logarithmic ...
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12 votes
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If $\Phi\geq 0$ is non-decreasing, does $\int_1^\infty \frac{\Phi(x)}{x^2}\,dx=\infty$ imply $\int_0^\infty e^{-\Phi(x)}\,dx<\infty$?

We may assume $\Phi(1)=0$. Let $\Phi(x)=\int_1^x f(t) dt$ for some nonnegative function $f$. By assumption, we have $$ \int_1^{\infty} \int_1^x \frac{f(t)}{x^2}dtdx =\int_1^{\infty}\int_t^{\infty} \...
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  • 17.8k
12 votes

$e^{-nx}\cdot\sum_{k=0}^\infty\frac{(nx)^k}{k!}f\left(\frac{k}{n}\right)\to f(x)$ for $f$ continuous and bounded

There's a nice one-liner proof using probability theory, since you effectively are looking at $E[f(X/n)]$ where $X$ is Poisson($nx$) distributed. A purely real analysis argument which is really based ...
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  • 94.3k
12 votes
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Rank 1 operator on an infinite dimensional vector space number of eigenvalues.

Yes, in general, this is true: the rank is an upper bound on the number of non-zero eigenvalues. If we have a linear operator $T : X \to X$, where $X$ is not necessarily finite-dimensional, then ...
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  • 43.3k
11 votes

Estimating the value of $e$ using a random function

Draw $X_i \sim \text{Unif}(1,3)$ and $Y_i \sim \text{Unif}(0,1)$ for $i=1,\ldots,N$. Reject the samples where $Y_i > 1/X_i$. Sort the accepted $X_i$ values and take the $(N/2)$th. This relies on ...
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11 votes
Accepted

Can there be a $C^1$ everywhere but $C^2$ nowhere function?

Take a continuous function which is differentiable nowhere, such as the Weierstrass function. Then take an antiderivative of that function.
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11 votes
Accepted

Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

A Couple of Trigonometric Sums First, we evaluate $$\newcommand{\Re}{\operatorname{Re}}\newcommand{\Im}{\operatorname{Im}} \begin{align} S_n &=\sum_{k=1}^n\sin(k)\tag{1a}\\ &=\Im\left(\frac{e^{...
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  • 327k
10 votes

Are there extensions of Euler's infinite product for sine function?

The generalizations of Euler's infinite product formula for the sine are applications of the Weierstrass factorization theorem: $$\prod_{n=1}^\infty \left( 1- \left( \frac{x}{n\pi + a}\right)^2\right)=...
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10 votes
Accepted

Prove $\int_{0}^{\infty} \psi^{(2)} (1+x) \ln (x) \, dx = \zeta'(2) + \zeta(2)$

Using the representation $$\psi^{(2)}(x) = - 2 \sum_{n=0}^{\infty} \frac{1}{(n+x)^{3}},$$ we have $$ \begin{align} \int_{0}^{\infty} \psi^{(2)}(x+1) \ln (x) \, \mathrm dx &= -2\int_{0}^{\infty}\ln(...
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10 votes
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Convergence of $\sum_{n=1}^{\infty}\frac{(\frac{2}{3}+\frac{1}{3}\cdot \sin(n))^n}{n}$

I summarize below the main ideas of the paper [1] whose link is in my comment above: Group the terms in the sum into "tame" and "wild" terms. The tame are defined as intergers ...
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  • 2,145
10 votes
Accepted

Show that $\frac{1}{n}\sum_{j=1}^\infty \left( 1 - (1-p_j)^n\right) \to 0$ as $n \to \infty$

I think your bound suffices! In fact, we have that $$ \frac{1}{n} \sum_{j=k+1}^\infty (1-(1-p_j)^n) \leq \frac{1}{n} \sum_{j=k+1}^\infty n p_j = \sum_{j=k+1}^\infty p_j. $$ This way, for each $k$ we ...
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10 votes
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Can you prove that these two series are equal?

Apply partial fraction expansion to the $n^{th}$ term of the series of $f(x)$: $$ \frac{1}{x \cdot (x+1) \cdots (x+n)} = \frac{A_0}{x} + \frac{A_1}{x+1} + \cdots + \frac{A_n}{x+n} $$ The coefficients ...
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10 votes
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Relation of Hamel basis with the equation $f(x + y) = f(x) + f(y)$?

The connection is the following one. A linear function from $\mathbb{R}$ to $\mathbb{R}$, where $\mathbb{R}$ is considered as a vector space over itself, is a function $f$ that satisfies the following ...
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  • 4,803
9 votes

Convergence of $\sum_{n=0}^{\infty} \frac{4^n}{3^n+7^n}$

Comparison test: Observe that $$\frac{4^n}{3^n+7^n} \le \frac{4^n}{7^n} = \left(\frac47\right)^n$$ which is a convergent geometric series Root test: Observe that $$\frac{4}{\sqrt[n]{7^n+7^n}}\le \...
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  • 11.5k
9 votes
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Can you string together inequalities and equalities into a single statement?

I've seen and used that sort of "stringing" together quite frequently, and to my understanding it is well-understood what it means. There shouldn't be any issue with using it. Whether it's &...
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9 votes

General form for $\sum_{n=1}^{\infty} (-1)^n \left(m n \, \text{arccoth} \, (m n) - 1\right)$

I didn't want to rain on mathstackuser12's parade, but now that the bounty has been awarded, here's a simpler approach using the theory of the Barnes G function. (Mathstackuser12 seems to have ...
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  • 6,503
9 votes
Accepted

To prove that the set of all polynomials with coefficient of $x^2$ equal to $0$ is dense in $C([0,1])$.

Let $\epsilon >0$. $g(x)=f(x^{1/3})$ defines a continuous function, so there is a polynomial $p$ such that $|f(x^{1/3})-p(x)|<\epsilon$ for all $x$. Hence, $|f(x)-p(x^{3})| <\epsilon$ for ...
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9 votes

$\int_a^bf'=f(b)-f(a)$ if $f'$ is integrable, but not continuous?

There are a few versions of the FTC, which depend on the hypotheses you decide to impose. All of these can be found in Rudin's RCA chapter $7$. For the version you're asking about, note that Rudin's ...
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  • 36.4k
9 votes
Accepted

Is this series known?

Since $a$ can only contribute nontrivially when even, we may as well replace $a\to 2a$ $$\sum_{(a,b)\in\Bbb{Z}^+\times\Bbb{Z}^+} \frac{2}{8a^3-4a^2+(2a+1)b^2} = \sum_{a=1}^\infty \frac{2}{2a+1}\sum_{b=...
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  • 25.8k
9 votes
Accepted

If every metric on a set X is equivalent to the discrete metric then can we say X is a finite set?

Assuming (a weak form of) the axiom of choice, yes, this is true. Specifically, from the axiom of choice it follows that if $X$ is infinite then there is an injection $f:\mathbb{Q}\rightarrow X$ (of ...
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