173 votes

Pointwise vs. Uniform Convergence

Comparison Pointwise convergence means at every point the sequence of functions has its own speed of convergence (that can be very fast at some points and very very very very slow at others). ...
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169 votes
Accepted

How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?

$\phantom{}$ Dear MSE users, this is the new episode of Mister Feynman and Monsieur Laplace versus contour integration. Tonight we have a scary integral, but we may immediately notice that $$ \...
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167 votes
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A continuous, nowhere differentiable but invertible function?

Interestingly, there are no such examples! For a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ to be invertible, it must be either monotone increasing or decreasing. A famous classical ...
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  • 4,403
140 votes

Why is the construction of the real numbers important?

First of all, mathematics is based on intuition and on concrete (imaginary, but concrete) objects, often inspired by reality. Let's stop the formalities for a moment and speak freely: we don't think ...
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  • 26.3k
105 votes

What is integration by parts, really?

I've always found it helpful to think about it like this: (picture source) The area of the gray areas combined is $u_2v_2 - u_1 v_1$, which is where the $uv$ term comes from.
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100 votes
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"Gaps" or "holes" in rational number system

It depends on what you consider a “gap” in the rational numbers. As long as this is not a formally defined concept, we’re just talking about our everyday, geometrically informed ...
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  • 216k
95 votes
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What is the theme of analysis?

I think that I'd say that one of the underlying themes of analysis is, really, the limit. In pretty much every subfield of analysis, we spend a lot of time trying to control the size of certain ...
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  • 11.6k
89 votes
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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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86 votes
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Are there periodic functions without a smallest period?

For a nontrivial example, consider the Dirichlet function, which has $$\delta(x) = \begin{cases}0 & \text{ if $x$ is rational}\\1 & \text{ if $x$ is irrational}\end{cases}$$ Then $\delta(x)$ ...
84 votes

Are there periodic functions without a smallest period?

Yes, for example constant function.
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78 votes

How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?

$\newcommand{\Res}{\text{Res}}$ Firstly define: \begin{align} f(z) = \frac{1}{z^2+bz+1} \end{align} Secondly define: \begin{align} g(z)= (-z)^{-ia}f(z) \end{align} We use the Principal Log to define $(...
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  • 8,438
69 votes

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

Of course it is a ratio. $dy$ and $dx$ are differentials. Thus they act on tangent vectors, not on points. That is, they are functions on the tangent manifold that are linear on each fiber. On the ...
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58 votes

A continuous, nowhere differentiable but invertible function?

Invertible implies bijective by set theory, and bijective together with continuity implies strictly increasing or decreasing, which imply differentiability almost everywhere! (This is known as ...
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54 votes
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A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$

Let the considered integral be denoted by $I$. Our starting point is to reduce the number of logarithms of different arguments in the integrand. Thus, using the fact that $6ab^2=(a+b)^3-2a^3+(a-b)^3$ ...
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54 votes
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Why doesn't the nested interval theorem hold for open intervals?

Consider the family $A_n = (0, \frac{1}{n})$. We have $A_{n+1} \subset A_n$ for every $n \in \mathbb{N}$, and the length of $A_n$ approaches zero as $n$ approaches infinity, but $\bigcap_{n=1}^{\infty}...
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50 votes

"Gaps" or "holes" in rational number system

There is a difference between a thing not existing in some set, and the existence of "gap" corresponding to that thing. For example, there is no rational number $p$ such that $p > q$ for all ...
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  • 24.1k
49 votes
Accepted

Baby/Papa/Mama/Big Rudin

In order to sum up the above comments, the corresponding books for these nick names are $1$. Baby = Principles of Mathematical Analysis; $2$. Papa/Big = Real and Complex Analysis; $3$. Grandpa = ...
49 votes

Why does Rudin say "the rational number system is inadequate as a field"?

The author means that $\mathbb Q$ as an ordered field is incomplete, i.e. not every Cauchy sequence in $\mathbb Q$ converges in $\mathbb Q$, or equivalently not every nonempty subset of $\mathbb Q$ ...
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  • 4,043
47 votes

What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?

For given $n\geq2$ one has $$e\cdot n!=n!\sum_{k=0}^\infty{1\over k!}=n!\left(\sum_{k=0}^n{1\over k!}+\sum_{k=n+1}^\infty{1\over k!}\right)=m_n+r_n$$ with $m_n\in{\mathbb Z}$ and $${1\over n+1}<r_n=...
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46 votes

Addition is to Integration as Multiplication is to ______

If $f\colon [a,b] \to (0,\infty)$, then the closest analogue to a multiplicative integral of $f$ is $$ {\prod}_a^b \,f(x)^{dx} \;\underset{\mathrm{def}}{=}\; \exp\left(\int_a^b \ln f(x)\,dx\right). $$ ...
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  • 46.5k
44 votes

What is integration by parts, really?

Integration by parts is a corollary of the product rule: $(uv)' = uv' + u'v$ Take the integral of both sides to get $uv = \int u \ dv + \int v \ du$. If you were supposed to remember it separately ...
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44 votes

The staircase paradox, or why $\pi\ne4$

Intuitive Response (for those who don't understand the more analytical responses) The answer is easy. We just have to zoom in. We can see at low zoom how the (purple) staircase hugs circle, but ...
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  • 956
44 votes

Why is the notion of analytic function so important?

Analytic functions have several nice properties, including but not limited to: They are $C^\infty$ functions. If, near $x_0$, we have$$f(x)=a_0+a_1(x-x_0)+a_2(x-x_0)^2+a_3(x-x_0)^3+\cdots,$$then$$f'...
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43 votes

Why is the construction of the real numbers important?

Imagine a mathematical paper that starts with: "Let $\mathbb G$ be a countable complete ordered field." We know that such an object doesn't exist, so anything that the paper has to say about it is ...
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  • 60.3k
42 votes

How to know if a term is divisible by 10

Your expression can be written as $16^{500}-6^{500}$. Recall that $a^n-b^n$ is divisible by $a-b$ for $n \in \mathbb{N}$.
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  • 5,321
42 votes
Accepted

Meaning of the backslash operator on sets

It’s set theoretic complement and in this case it denotes the set of all reals which are not natural: $$ℝ \setminus ℕ = \{x ∈ ℝ;~x \notin ℕ\}$$
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  • 17.7k
42 votes

Why does $1+2+3+\cdots = -\frac{1}{12}$?

This infinite series is ultimately divergent because $$ 1+2+3+4+\cdots=\sum\limits_{k=1}^{\infty} k$$ $$ = \lim\limits_{n\to\infty} \sum\limits_{k=1}^{n} k = \lim\limits_{n\to\infty} \frac{n(n+1)}{2} ...
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  • 8,591

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