Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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68
votes
8answers
12k views

Why does $1+2+3+\dots = {-1\over 12}$?

$$\lim_{N\to\infty} \sum_{i=1}^N \, i = +\infty$$ $\displaystyle\sum_{n=1}^\infty n^{-s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ ...
223
votes
14answers
18k views

Is value of $\pi = 4$?

What is wrong with this? SOURCE
57
votes
18answers
11k views

Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_{0}^{\infty} \frac{\sin x}{x} \ dx = \frac{\pi}{2}$$ Well, can anyone ...
22
votes
8answers
4k views

Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}.$

I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}.$$ What do you think? It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} dx$$ is also ...
12
votes
4answers
2k views

Sine function dense in $[-1,1]$

We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by ...
54
votes
4answers
3k views

Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
6
votes
8answers
531 views

How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$?

I've spent the better part of this day trying to show from first principles that this sequence tends to 1. Could anyone give me an idea of how I can approach this problem? $$ \lim_{n \to +\infty} ...
5
votes
1answer
677 views

Inequality involving $\limsup$ and $\liminf$

This may have been asked before, however I was unable to find any duplicate. This comes from pg. 52 of "Mathematical Analysis: An Introduction" by Browder. Problem 14: If $(a_n)$ is a sequence in ...
12
votes
3answers
1k views

Upper and Lower Bounds of $\emptyset$

From some reading, I've noticed that $\sup(\emptyset)=\min(S)$, but $\inf(\emptyset)=\max(S)$, given that $\min(S)$ and $\max(S)$ exist, where $S$ is the universe in which one is working. Is there ...
22
votes
1answer
2k views

Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$

Here is an exercise, on analysis which i am stuck. How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on ...
8
votes
2answers
1k views

How do you show that $l_p \subset l_q$ for $p \leq q$?

I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
5
votes
3answers
546 views

Combinatorial proof

Using notion of derivative of functions from Taylor formula follow that $$e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ Is there any elementary combinatorial proof of this formula here is my proof for ...
11
votes
5answers
2k views

Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$

As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem ...
1
vote
3answers
412 views

Prove that a function having a derivative bounded by 0.49 has a unique solution $\frac{2x+\sin(x)}{2}$

Let f : $\mathbb{R} \to \mathbb{R}$ be differentiable function and suppose that $|f'(x)|\le 0.49$ for all $x \in \mathbb{R}$. Prove that the equation $f(x) =\frac{2x+\sin(x)}{2}$ has a unique ...
18
votes
7answers
4k views

Any open subset of $\Bbb R$ is a countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
21
votes
1answer
2k views

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

I tried and got this $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an ...
14
votes
3answers
1k views

$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

Trying to solve $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges. I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$ Would ...
22
votes
2answers
1k views

A stronger version of discrete “Liouville's theorem”

If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, ...
13
votes
8answers
4k views

Why Circle encloses largest Area?

In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would ...
3
votes
4answers
185 views

A Convergent Series

How to prove the following equality? $$\sum_{n=1}^{\infty}\frac{1}{\prod_{k=1}^{m}(n+k)}=\frac{1}{(m-1)m!}.$$
7
votes
5answers
656 views

How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$?

I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it ...
14
votes
3answers
661 views

Choice of $q$ in Baby Rudin's Example 1.1

First, my apologies if this has already been asked/answered. I wasn't able to find this question via search. My question comes from Rudin's "Princicples of Mathematical Analysis," or "Baby Rudin," ...
17
votes
2answers
857 views

Continuity and the Axiom of Choice

In my introductory Analysis course, we learned two definitions of continuity. $(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ ...
14
votes
2answers
657 views

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere

Prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y$, and $f$ is continuous at $0$, then it is continuous everywhere. If there exists $c \in \mathbb{R}$ ...
6
votes
5answers
2k views

If $f'$ tends to a positive limit as $x$ approaches infinity, then $f$ approaches infinity

Some time ago, I've asked this here. A restricted form of the second question could be this: If $f$ is a function with first derivative continuous in $\mathbb{R}$ and such that $$\lim_{x\to \infty} ...
5
votes
2answers
311 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} ...
15
votes
2answers
750 views

If $f,g$ are both analytic and $f(z) = g(z)$ for uncountably many $z$, is it true that $f = g$?

If two analytical functions of $\mathbb{C}$ f and g are equal on an infinite number of input values, than they are equal. I can't seem to find a counterexample, but I haven't seen this anywhere ...
4
votes
2answers
612 views

Uniform continuity

I need to prove that if $f: (0,1) \rightarrow \mathbb{R}$ is Uniformly continuous then it is bounded. Thank you.
18
votes
3answers
1k views

Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$ Computing the Fourier series of $f$ and ...
17
votes
4answers
2k views

Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
12
votes
1answer
673 views

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic Maybe I would have to use the Rademachers.
11
votes
3answers
733 views

Is the Ratio of Associative Binary Operations to All Binary Operations on a Set of $n$ Elements Generally Small?

I started thinking about the number of associative (binary) operations on a set with $n$ elements today. Looking online I found this paper which indicates only $113$ of the possible $19,683$ ...
1
vote
6answers
4k views

Show that a continuous function has a fixed point

Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$. I suppose this has ...
1
vote
2answers
437 views

$\limsup $ and $\liminf$ of a sequence of subsets relative to a topology

From Wikipedia if $\{A_n\}$ is a sequence of subsets of a topological space $X$, then: $\limsup A_n$, which is also called the outer limit, consists of those elements which are limits of ...
49
votes
3answers
3k views

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}dx$

I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( ...
13
votes
1answer
723 views

Compute the inverse Laplace transform of $e^{-\sqrt{z}}$

I want to compute the inverse Laplace transform of a function $$ F(z) = e^{-\sqrt{z}}. $$ This problem seems very nontrivial to me. Here one can find the answer: the inverse Laplace transform of ...
21
votes
2answers
803 views

Global invertibility of a map $\mathbb{R}^n\to \mathbb{R}^n$ from everywhere local invertibility

I was told by a tutor that if $f: \mathbb{R}^n \longrightarrow \mathbb{R}^n$ has an invertible Jacobian Matrix for all $x \in \mathbb{R}^n$ and $\lim_{|x_k| \rightarrow \infty}|f(x_k)|=\infty$ for all ...
16
votes
4answers
687 views

Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?

Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere? I think it is probable because we can consider $$ y ...
6
votes
3answers
517 views

Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$

How does one sum the series, $$ S = a -\frac{2}{3}a^{3} + \frac{2 \cdot 4}{3 \cdot 5} a^{5} - \frac{ 2 \cdot 4 \cdot 6}{ 3 \cdot 5 \cdot 7}a^{7} + \cdots $$ This was asked to me by a high school ...
3
votes
6answers
942 views

Bernoulli's representation of Euler's number, i.e $e=\lim \limits_{x\to \infty} \left(1+\frac{1}{x}\right)^x $ [duplicate]

Possible Duplicates: Finding the limit of $n/\sqrt[n]{n!}$ How come such different methods result in the same number, $e$? I've seen this formula several thousand times: $$e=\lim_{x\to ...
2
votes
2answers
232 views

prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.

Let $f$ be a real-valued function continuous on $[a,b]$ and differentiable on $(a,b)$. Suppose that $\lim_{x\rightarrow a}f'(x)$ exists. Then, prove that $f$ is differentiable at $a$ and ...
7
votes
6answers
767 views

Defining division by zero [duplicate]

I have looked through some of the previous questions posted on this topic, and I think mine is different. Is there a flaw in defining division by zero? For example, define $\frac{a}{0} = \infty_a$ ...
6
votes
2answers
618 views

Asymptotic formula for $\sum_{n\leq x}\mu(n)[x/n]^2$ and the Totient summatory function $\sum_{n\leq x} \phi(n)$

I would like to show (for $x \ge 2$) that $$\sum_{n \le x}\mu(n)\left[\frac{x}{n}\right]^2 = \frac{x^2}{\zeta(2)} + O(x \log(x)).$$ I already have the identity $$\sum_{n \le ...
2
votes
2answers
184 views

prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:

$T(a)=\begin{cases} 0 & a\leq 0\\ e^{-1/a} & a>0 \end{cases}$ $Z(a)=\begin{cases}0 & a\geq 1\\ (1+a) e^{-1/(1-a)} & a<1 \end{cases}$ $p=\int_{-1}^{1}Z(x)dx$ ...
18
votes
2answers
554 views

Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
16
votes
4answers
673 views

Evaluation of the integral $\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$

How can I evaluate the integral $$\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$$ I tried manipulating the known integral $$\int_0^1 \frac{\ln(1 - x)}{x}dx = -\frac{\pi^2}{6}$$ but couldn't do anything with ...
17
votes
1answer
852 views

A smooth function's domain of being non-analytic

I am wondering how much a smooth function may be non-analytic, because in proofs, whilst there non-analytic smooth functions, it would suffice if a smooth function were analytic on only a "small set". ...
10
votes
3answers
515 views

How does Lambert's W behave near ∞?

How does $W$ behave near $+\infty$ compared to $\log$? In particular, I'm interested in the asymptotic expansion of $$\frac{W(x)}{\ln(x)}$$ near $\infty$ (but along the positive real line, if that ...
10
votes
5answers
580 views

Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$

If $x \geq 2$, then how do we prove that $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
7
votes
1answer
477 views

Tensor products of functions generate dense subspace?

Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...