Questions about studying mathematics without formal instruction.

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2
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1answer
84 views

exam exercise on Series problem.

The exercise states: Does the series $$\sum\limits_{n=1}^\infty \int_{0}^1 \frac{x^n dx}{x+1}$$ converge? The solution states as the first step: $$I_n =\int_{0}^1 \frac{x^n dx}{x+1} $$ then $ ...
2
votes
1answer
38 views

Testing convergence of the series of $n^p((n-1)^{-1/2}-n^{-1/2})$

Exercise 8.15 (l) of Analysis by Apostol states: Test for convergence: $$\sum\limits_{n = 2}^\infty n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt{n}} \right)$$ The solution I have states as the first ...
1
vote
2answers
81 views

Books/subjects for proof practice

So I want to practice writing proofs. I've studied general proof-writing but now I want to learn how to apply that to mathematics. From what I understand, the best and most accessible subjects for ...
2
votes
1answer
117 views

$[0,1]^{[0,1]}$ is separable

This is from Dudley´s book: Let $I:=[0,1]$ with the usual topology. Let $I^I$the set of all the functions from $I$ to $I$ with the product topology. a) $I^I$ is separable. Hint: Consider function ...
0
votes
1answer
29 views

Showing a set is closed, question from real analysis

Let $\mathbb{X}=\{1,2,3\}$ and let $\mathbb{P}$ be the set of all probabilities on $\mathbb{X}$. $\,\,$ Let, $V:\mathbb{P}\to \mathbb{R}$ be defined as $V(p)=(1+p_1)^2+(2+p_2)^2+(3+p_3)^2$. Show that ...
1
vote
2answers
32 views

How many ways there are to divide $5$-element set to at mot three sets?

How many ways there are to divide $5$-element set to at most three subsets? If I am right, then I have following subsets: 1 subset: containing five elements (that is 1 possibility) 2 subsets: 1 ...
3
votes
1answer
36 views

Find inverse of $I+\mathbf{ab}^\intercal$

Could you guys give me some hints on this homework? Find inverse of $\mathbf{I} + \mathbf{ab}^\intercal$. Hint: try to form $c\mathbf{I} + d\mathbf{ab}^\intercal$ and solve for $c,d$. What happens ...
0
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3answers
15 views

Simple Conditional Variance Proof Question

So I have the following best linear predictor: $y_{t+1} = a + b y_t + v_t $ , where $b$ is a measure of persistence and $v_t$ is noise (independent of $y_t$). Variance is persistent across ...
9
votes
4answers
148 views

Is there a closed form expression for $\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )} \mathrm{d}x\,\mathrm{d}y$?

I have been trying to evaluate the integral: $$\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )}\mathrm {d}x\,\mathrm{d}y$$ I know of course that the integral equals ...
3
votes
1answer
63 views

Introductory texts in abstract algebra, and game theory taking non-standard approaches

I like to see subjects from different angles. For example in linear algebra I'm reading through Axler's text (which takes a proof based approach for math students), but I'm also checking out a text on ...
1
vote
1answer
97 views

What is the (rigorous) reason that the derivative of $|x|$ does not exist at $x=0$?

Let $g=|x|$. Then, the derivative at $c=0$ is given by: $$ g'(0) = \lim_{x \to 0} \frac{|x|}{x} $$ which is either $+1$ if $x$ comes from the positive $x$-axis or $-1$ if $x$ comes from the negative ...
2
votes
0answers
91 views

A Compact Real Analysis book for a graduate student, who is short of time.

I am a Phd student in Computer Science and I want to focus on Machine Learning, especially on statistical methods. My problem is, I always keep hitting the wall when it comes to studying underlying ...
1
vote
1answer
26 views

Proof check from basic set theory

I wonder if my proof is detailed enough. $f:X\to Y.$ To be proved: $f^{-1}(\bigcap_{\alpha}E{\alpha})=\bigcap_{\alpha}(f^{-1}E_{\alpha}) $ $$\,$$ So, here goes ...
0
votes
1answer
47 views

On complex numbers and absolute values

Exercise 1.31 of Analysis by Apostol states: Given three complex numbers $z_1,z_2,z_3$ such that $|z_1| = |z_2| = |z_3| = 1$ and $z_1 + z_2 +z_3 = 0$. Show that these numbers are vertices of an ...
0
votes
1answer
29 views

Simplification Problem

$$\left | \frac{4-3m_3}{3+4m_3} \right |= \left | \frac{-3-4m_3}{4-3m_3} \right |$$ I am always confused when it comes to modulus. I know if there is modulus any one of the side then when we remove ...
1
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0answers
58 views

Suggested book for self study.

I have a degree in Financial Risk Management, and did 4 semesters of calculus and analysis(but that was about 10 years back), with most of my other efforts going toward Mathematical Statistics and ...
1
vote
2answers
51 views

One Question on Law of Total Probability

Let $(X_n)$ with $n \in \mathbb N_0$ be a discrete martingale. Then I read the following identity which is said to be derived from the law of total probability. $$ \mathbb EX_m = \left( \sum_{n=0}^m ...
1
vote
1answer
29 views

Intuition underlying stopped martingales

Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is ...
1
vote
0answers
25 views

net of indicator functions

Hi everyone I was reading Dudley's book and I'm having problems with the following. If $X$ is uncountable, show that there is a net of indicator functions of finite set converging pointwise to the ...
2
votes
5answers
120 views

What is negation of the following sentence?

What is negation of All birds can fly. The question seems bit funny but i don't know which of the following two sentences is correct: Some birds can not fly There is at least one bird which can not ...
0
votes
0answers
35 views

Can I start apostol (vol 2) with no background in multivariable calculus?

I searched a lot in web and almost everyone says if you want to read spivak or apostol, you should first read an introductory book on calculus like stewart. I didn't read stewart but I studied single ...
1
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0answers
81 views

Learning math vs problem solving

Ok so I am about to start my final year in high school we will be learning calculus this year, but I already know single- and multi-variable calculus and linear algebra so I want to spend my final ...
1
vote
1answer
57 views

Question about limit points in relation with continuity and functional limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have the feeling that the author is being careless about limit points in his theorems or I am not understanding ...
1
vote
2answers
65 views

Proof of Divergence Criterion for Functional Limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I don't understand corollary 4.2.5 on page 107. To be more specific, let me first write down the theorem that precedes ...
1
vote
4answers
41 views

Find $n(A \cap B)$

Question: In group of people, 60% like coffee and 70% like tea. How many people like both of them.? My Effort: We have to find how many people like both the items that means we have to find $n(A ...
1
vote
1answer
46 views

on limits of cumulative density functions

Theorem 1.5.3 of Statistical Inference by Casella and Berger States that the function $F(x)$ is a cdf if and only if the following three conditions hold: 1) $\lim_{x \to - \infty} F(x) = 0 \text{ and ...
34
votes
5answers
1k views

Self-learning mathematics - help needed!

First, I apologise for the nebulous nature of my title but I can't adequately explain myself concisely. I am about to start an MSc in pure maths after a fairly shaky undergraduate degree. I am very ...
3
votes
0answers
57 views

How to approach real analaysis

I'm just starting first year in university in Europe and here there there is no Calculus, instead you jump right into Analysis. The trouble is, for some time I self-studied through US style books and ...
0
votes
1answer
14 views

Find catesian coordinate of T-point $P(-\frac{65\pi}{2}) $

Find the Cartesian coordinates of T-point $P(-\frac{65\pi}{2}) $. It is easy when there is no negative sign. I don' t know how to do with negative sign.
1
vote
1answer
32 views

Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
2
votes
0answers
51 views

What is the big picture of mathematics? [closed]

I learn from theoretical to concrete - i see the forest Then the trees. It seems conventional studying of mathematics take you through by Showing you tree after tree til you have a forest. Big ...
3
votes
2answers
133 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
1
vote
2answers
24 views

Finding Principale period of $\cos$ function

Find principle period of $3\cos (2x-3)$. Today I have learned about principle period of various trigonometric function. I know that principle period of cos is $2 \pi$. Please someone can help me ...
0
votes
1answer
22 views

prove that $g$ is a function of $(x_1-x_2,x_2-x_3,\dots,x_{n-1}-x_n)$

$g$ is a function of $(x_1-x_2,x_2-x_3,\dots,x_{n-1}-x_n)$ iff $$g(x_1+a,x_2+a,\dots,x_n+a)=g(x_1,x_2,\dots,x_n)\forall a \in \mathbb R$$ Trial: Only if part : Consider $g$ is a function of ...
0
votes
1answer
35 views

How to get this inequality

Let $c>0$, $n \in \mathbb N$ and $q>1$. How to get the following approximating inequality when $n$ is large, please? To be more specific, I cannot see how to get rid of the square root. $$ ...
0
votes
1answer
50 views

Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...
2
votes
2answers
40 views

exercise on pointwise convergence of an (easy) function.

Exercise 6.2.5. Taken from understanding analysis of Stephen Abbott For each n $\in N$, define $f_n on \ R$ by $$f_n(x) = \begin{cases} 1, & \mbox{if} \ |x| \ge 1/n \\ n|x|, & \mbox{if} \ ...
2
votes
3answers
101 views

If $G$ is a simple $f$ an homomorphism, and $A\lhd H$ is such that $[H:A]=2$, show $f(G) \subset A$

I'm stuck with the following problem. Can someone help me by providing a hint? Suppose $G$ is simple and let $f$ be an homomorphism between $G \to H$. If $\#G\ne2$, $A\lhd H$, and $[H:A]=2$. Then ...
4
votes
1answer
62 views

hint with an exercise algebra

I'm stuck with the following I hope someone could help me. Let $N$ a normal subgroup of $G$. Show that if $[G:N]=4$, exists a normal subgroup $M$ of $G$ s.t. $[G:M]=2$. My idea: Since $G/N$ has ...
1
vote
1answer
39 views

Prove that line segments are parallel.

Prove using slope of lines that line segment joining the midpoint of $\overline { AB}$ and $\overline{AC}$ in $\Delta ABC$ is parallel to $\overline {BC.}$ Need to prove using slope of lines means I ...
2
votes
5answers
797 views

How should I self-study calculus?

So I already took Pre-Calc, and ended up with a B both semesters. I am an incoming senior in high school. My special-ed case manager won't let me take it because she doesn't want to see me panic ...
0
votes
0answers
32 views

What would be the way to learn algorithms?

I want to know the mathematical theory associated algorithms. Story with a base of calculation, analytical geometry and linear algebra. What would be the previous topics to begin the study of ...
1
vote
2answers
128 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
0
votes
1answer
48 views

General formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Is such proof correct?

I have a very simple case: Find general formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Of course dividing one by another was quite simple with outcome: $a^n(a+1)$. However I would like to prove ...
0
votes
1answer
74 views

Monotonic Functions and Uniform Convergence

The following is a proof from "Heavy-Tail Phenomena" by Resnick (2007). I have some questions about the proof. (2.3) seems to be an identity. The left side the global sup over $[a, b]$ and hence ...
3
votes
3answers
116 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
2
votes
1answer
40 views

Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
4
votes
2answers
162 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
2
votes
1answer
28 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
2
votes
1answer
36 views

Using Taylors to show convergence in probability

I'd like to show that \begin{equation} \sqrt{n} \left( (1-\frac{1}{n})^{n\bar{X}} - e^{-\bar{X}} \right) \to 0 \end{equation} in probability for a random variable with mean $\mu$ and finite variance ...