240

R.I.P. Archimedes A photogenic answer to such a question!


199

there are many ways to see that your result is the right one. What does the right one mean? It means that whenever such a sum appears anywhere in physics - I explicitly emphasize that not just in string theory, also in experimentally doable measurements of the Casimir force (between parallel metals resulting from quantized standing electromagnetic waves in ...


167

$\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}$I will evaluate the sum at the end of Jim Belk's post. There is clearly a lot more going on here; I hope a number theorist will come and explain what is really going on. I will be sloppy about convergence issues. Set $$S:= \sum_{k,n=1}^{\infty} \frac{(-1)^{k+n}}{k^2+4kn+n^2}.$$ We set $m = k+2n$, in order to ...


157

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 result in analysis, Lebesgue's Monotone Function Theorem, states that any monotone function on an open interval is differentiable almost everywhere. Hence, there ...


142

$\phantom{}$ 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 $$ \mathcal{L}(x^{-ia})(s) = s^{ia-1}\Gamma(1-ia),\qquad \mathcal{L}^{-1}\left(\frac{1}{x^2+bx+1}\right)(s) =\frac{e^{Bs}-e^{\overline{B}s}}{\sqrt{b^2-4}}$$ where $B$ is ...


125

As many have said, compactness is sort of a topological generalization of finiteness. And this is true in a deep sense, because topology deals with open sets, and this means that we often "care about how something behaves on an open set", and for compact spaces this means that there are only finitely many possible behaviors. But why finiteness is important? ...


124

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 of $3/4+1/2$ as an operation involving equivalence classes of ordered pairs of Von Neumann integers, we think of it as pouring $3/4$ liters of water and $1/2$ ...


121

My favorite "counterexample" to the derivative acting like a ratio: the implicit differentiation formula for two variables. We have $$\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y} $$ The formula is almost what you would expect, except for that pesky minus sign. See http://en.wikipedia.org/wiki/Implicit_differentiation#...


117

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). Imagine how slow that sequence tends to zero at more and more outer points: $$\frac{1}{n}x^2\to 0$$ Uniform convergence means there is an overall speed of ...


111

I like to remember this as a special case of the Fubini/Tonelli theorems, where the measures are counting measure on $\mathbb{N}$ and Lebesgue measure on $\mathbb{R}$ (or $[0,\infty)$ as you've written it here). In particular, Tonelli's theorem says if $f_n(x) \ge 0$ for all $n,x$, then $$\sum \int f_n(x) dx = \int \sum f_n(x) dx$$ without any further ...


110

Here is a variant on Lubos Motl's answer: Let $S = \sum_{n=1}^{\infty} n$. Then $S - 4 S = \sum_{n = 1}^{\infty} (-1)^{n-1} n.$ We will evaluate this latter expression with a regularization similar to Lubos Motl's. Namely, consider $$\sum_{n=1}^{\infty} (-1)^{n-1} n t^n = -t \dfrac{d}{dt} \sum_{n=1}^{\infty} (-t)^n = -t \dfrac{d}{dt} \dfrac{1}{1+t} = \...


108

This isn't an answer to the question, but I thought I should post some of my work here. Consider the function $$ F(a) \;=\; \int_0^1 \frac{\log(1+x^a)}{1+x}dx. $$ The question asks us to prove that $F(2+\sqrt{3}) = \dfrac{\pi^2}{12}(1+\sqrt{3}) + \log(2)\log(1+\sqrt{3})$. Mathematica isn't able to compute a closed form for $F(2+\sqrt{3})$, but it can ...


105

Here’s one to get things started. Let $U$ be a non-empty open subset of $\Bbb R$. For $x,y\in U$ define $x\sim y$ iff $\big[\min\{x,y\},\max\{x,y\}\big]\subseteq U$. It’s easily checked that $\sim$ is an equivalence relation on $U$ whose equivalence classes are pairwise disjoint open intervals in $\Bbb R$. (The term interval here includes unbounded ...


93

I think this (interesting!) question is yet-another instance of the occasional mismatch of science (and human perception) and formal mathematics. For example, the arguments used by the questioner, and common throughout science and engineering, were also those used by Euler and other mathematicians for 150 years prior to Abel, Cauchy, Weierstrass, and others' ...


86

In Leibniz's mathematics, if $y=x^2$ then $\frac{dy}{dx}$ would be "equal" to $2x$, but the meaning of "equality" to Leibniz was not the same as it is to us. He emphasized repeatedly (for example in his 1695 response to Nieuwentijt) that he was working with a generalized notion of equality "up to" a negligible term. Also, Leibniz used several different ...


86

It may help if you unfold the definition of limit in pointwise convergence. Then pointwise convergence means that for each $x$ and $\epsilon$ you can find an $N$ such that (bla bla bla). Here the $N$ is allowed to depend both on $x$ and $\epsilon$. In uniform convergence the requirement is strengthened. Here for each $\epsilon$ you need to be able to find ...


84

I am probably going a little off-topic with these comments, so feel free to downvote :) In my opinion this type of proof emphasizes why it is wrong to teach/take “Calculus” instead of Analysis. For most of the nice applications of integration, we always use the following approach: take some quantity/expression, break it in many pieces, identify the sum of ...


84

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.


81

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 $\delta(x)$ is periodic with period $r$ for every rational number $r$.


80

What kind of algebraist "refuses to acknowledge the existence of any characteristic 0 field other than $\mathbb{Q}$"?? But there is a good question in here nevertheless: the basic definitions of limit, continuity, and differentiability all make sense for functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$. The real numbers are in many ways a much more ...


79

Yes, for example constant function.


74

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 quantities, with taking limits in mind. This is especially true in PDEs, when we consistently desire norm estimates on various quantities. Let's just discuss the "...


73

What a great method! I tried a few... $$ \sum_{n = 1}^{\infty} \left(\frac{1}{(2 n)^{s}} - \frac{1}{(2 n - 1)^{s}}\right) = \zeta (s) \Bigl(2^{(1 - s)} - 1\Bigr) $$ put $s=-1$ to get $\sum_{n = 1}^{\infty} 1 = -\frac{1}{4}$ or $$ \sum_{n = 1}^{\infty} \left(\frac{1}{(2 n + 1)^{s}} - \frac{1}{(2 n)^{s}}\right) = \zeta (s) - \frac{2 \zeta (s)}{2^{s}} - ...


73

You can use $\text{AM} \ge \text{GM}$. $$\frac{1 + 1 + \dots + 1 + \sqrt{n} + \sqrt{n}}{n} \ge n^{1/n} \ge 1$$ $$ 1 - \frac{2}{n} + \frac{2}{\sqrt{n}} \ge n^{1/n} \ge 1$$


70

$\newcommand{\Res}{\text{Res}}$ 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 $(-z)^{-ia}$. The main reason for the minus sign is that I want to work with the principal Log. Consider now the keyhole contour $K_R$ consisting of $C_R \cup ...


69

One method is to consider the generating function of $\zeta(2k)$: $$ \begin{align} f(x) &=\sum_{k=1}^\infty\zeta(2k)\,x^{2k}\\ &=\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{x^{2k}}{n^{2k}}\\ &=\sum_{n=1}^\infty\frac{x^2/n^2}{1-x^2/n^2}\\ &=\sum_{n=1}^\infty\frac{x^2}{n^2-x^2}\\ &=-\frac{x}{2}\sum_{n=1}^\infty\left(\frac{1}{x-n}+\frac{1}{x+n}\...


67

Let me ask you a dual question: I am a mathematics student in set theory, why don't category theory students do set theory? I find it strange that most books on category theory have only a naive handling of set theory. Now let me answer your question. Category theory is an impressive tool for abstraction, but analysis is not always in need for abstraction - ...


66

I believe that I may be of some consolation. I had a very similar experience to you. I started doing "serious" math when I was a senior in high school. I thought I was very smart because I was studying what I thought was advanced analysis--baby Rudin. My ego took a hit when I reached college and realized that while I had a knack for analysis and point-set ...


65

This web page has Theorem 6.1. It is written in Spanish, but actually is rather easy to follow even if (like me) you don't know any Spanish. However it is not made clear on this web site that the statement of Theorem 6.1 is "If $\|A^\theta \overset 0u\| \le C_\theta\|$, then $\| \overset0u \| \le C_1(1+\|\overset0f\|+\|\overset0f\|^l)$." http://francis....


Only top voted, non community-wiki answers of a minimum length are eligible