Point-wise continuity and convergence of limit function
Why doesn't the pointwise convergence of a sequence of continuous functions necessarily converge to a continuous limit function?
View ArticleIf $a_n\ge 0$ and $a_n\to 0$ and also $\sum a_n=\infty$, then $\sum...
Good Afternoon!am trying to understand a well known research paper, I came across the following lines, which let me assume that the conjecture is pretty "standard" or well known, however, I don't know...
View ArticleIn what way does Monotone Convergence imply LPO?
I’m studying the constructive proof that Limited Principle of Omniscience is equivalent to the Bolzano-Weierstrass theorem. The last step in the original Mandelkern’s proof is showing that the monotone...
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Newton's method to solve for $f=ax^2-x+1=0$
Let $\{x_k\}$ be sequence generated by Newton's Method for solving$ax^2-x+1=0$, where $0<a\leq\frac{1}{4}$. Suppose $x_0<\frac{1}{2a}$.Show...
View Articleprove that $ \sqrt{2} $ is irrational using using nested intervals
I want to prove $ \sqrt{2} $ is irrational using nested intervals.Assume, for contradiction, that $\sqrt{2}$ is rational. This means$$\sqrt{2} = \frac{a}{b}$$for some integers $a$ and $b$$(b /= 0)$,...
View ArticleDifference between finite partial sums from two divergent series
Fix a sequence $(r_i)_{i\in\mathbb{N}} \subseteq (0, 1)$ such that $\lim_i r_i=0$ and $\sum_{i\in \mathbb{N}} r_i=\infty$. According to the answer in this post, for any $c>0$ there exists...
View ArticleMeasurability of functions in the extended real numbers
I am trying to solve this problem: Let (X,S) be a measurable space, A ∈ S and f, g ∈ M(X,S ). The function h:S → [−∞,∞] defined as\begin{equation*} h(x) = \begin{cases} f(x) & \text{if } x \in A \\...
View ArticleQuasi-probability density functions and vanishing expectations
I am interested in the properties of quasi-probability density functions in two dimensions, namely, functions $\rho(x,y):\, \mathbb{R}^2\to \mathbb{R}$ that satisfy the...
View ArticleDoes the Lagrange Multipliers apply for this problem by ignoring some...
Problem. Given four real numbers$a_1 \ge a_2 \ge a_3 \ge a_4 > 0$such that $a_1 + a_2 + a_3 + a_4 = 9$and $a_1^2 + a_2^2 + a_3^2 + a_4^2 = 21$, find the minimum and maximum of...
View ArticleClosed form for $\sum_{n=1}^{\infty}\frac{n^k}{2n!}$
I'm looking if there's a general closed form for the following function:$$f(k) = \sum_{n=1}^{\infty}\frac{n^k}{(2n)!},$$for $k\in \Bbb{N}$.For specific $k$ values I can utilize the fact that$$\cosh(x)...
View ArticleProving $(\sum_{i=0}^{\infty}x^{i})^2=\sum_{n=0}^{\infty} (n+1)x^n$
We are trying to evaluate:$$\sum_{i=0,j=0}^{\infty}x^{i+j}=\sum_{n=0}^{\infty} (n+1)x^n$$My attempt:$$\sum_{i,j=0}^\infty x^{i+j}=\sum_{i=0}^\infty\sum_{j=0}^\infty x^{i+j}$$And obviously, the...
View ArticleConvergence or divergence of $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$
If $a_1=a_2=1$ and $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$ How to prove convergence of the sequence and its limit or divergence?It is easy to see that the sequence is always positive and by that one can...
View ArticleHypothesizing closed form of $(\sum_{n=0}^{\infty} x^n)^i$, where $i\ge 1$,...
In my previous question QuestionWe have arrived at the conclusion that $$(\sum_{n=0}^\infty x^n)^2=\sum_{n=0}^\infty (n+1)x^n$$And I found this infinite series can be written as...
View ArticleFinding series equation solution with analytic data
Consider the equation:$$ g(t) = \sum_{n=1}^{\infty} e^{-n^2 t} a_n, $$where $ g(t) $ is an analytic function on $[0,T]$ with $ g(0) = 0 $. I want to know if there exist $a_n$, $n\geq 1$ such that this...
View ArticleQuestion regarding the set of infinite series of {0,1} being uncountable
I've been learning the basics of real analysis lately and have been quite bothered by this question. After going through several proofs, I've been convinced by the proof making use of proper subsets,...
View ArticleSteinhaus theorem (sums version)
This is a question from Stromberg related to Steinhaus' Theorem:If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval.I can't quite see how to modify the Steinhaus proof...
View ArticleSelf-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges...
To prove the convergence of the p-series $$\sum_{n=1}^{\infty} \frac1{n^p}$$for $p > 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.I am wondering if there is...
View ArticleIs it possible for a two divergent series have a convergant difference?
I am working on a series where I have split it into 2.the first is the sum of a sequence $a_n$ that converge to $1/2$. and the second $b_n$ is the same thing. is it possible for their difference...
View ArticleLimit of diverging sequences.
I have a confusion in my mind. Consider two sequences of real numbers $\{x_n\}$ and $\{y_n\}$. Define $z_n: = x_n -y_n$. Suppose $x_n, y_n$ are two diverging sequences.Then what can be said about the...
View ArticleSolve $x^5 + x - 1 = 0$
Solve $x^5 +x - 1 = 0$I am simply curious to see how the solution would go, since it is a quintic; it cannot be done by regular methods.I'm just curious to see what people would come up with, and I can...
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