prove real analysis series question: $\sum^{\infty}_{i=0}a_i^2b_i^2 < \infty...
Suppose, $a_k$ and $b_k$ is non-negative sequences. $a_k$ decreasing, $\sum^{\infty}_{i=0}a_i $ diverge to $\infty$, and $\sum^{\infty}_{i=0}a_i^2$ converge.For $\sum^{\infty}_{i=0}a_i^2b_i^2 <...
View ArticleTwo representation of a particular form of hypergeometric function.
From the basic definition ofhypergeometric function,we know that$${}_{2}F_{1}(1,p;p+1;1) =p \int_{0}^{1} \dfrac{t^{p-1}}{1-t} dt. $$My professor also told me that\begin{align*}{}_{2}F_{1}(1, p;...
View ArticleMath Analysis Courses online
Can somebody recommend me respectable Math Analysis courses online?I am a student and I took real analysis course in my university, but I am unsatisfied with the quality of that course. I am even...
View ArticleFolland Theorem 2.15. How does this 'Let $N \to \infty$ and apply monotone...
I am reading Real analysis by Folland G.B. I have a slight doubt on this..Theorem 2.15If $(f_n)$ is a finite of infinite sequence in $L^+$ and $f=\sum_n f_n$, then $\int f = \sum_n \int f_n$In the...
View ArticleHow do I prove this function is bounded from below?
$A \in \mathcal{M}_{n,n}(\mathbb{R})$ a positive definite matrix, $b \in \mathbb{R}^n$ and $c \in \mathbb{R}. $$f : \mathbb{R}^n \to \mathbb{R}$ defined by : $$f(x) = \frac{1}{2}\langle Ax, x \rangle +...
View ArticleInequality involving finite number of nonnegative real numbers
If $0\leq c_k\leq 1$ and $n$ is any positive integer, then is$$\frac{n\prod_{k=1}^nc_k}{1+n\prod_{k=1}^nc_k}\leq \sum_{k=1}^n\frac{c_k}{1+c_k}?$$It is true for at least one of the $c_k=0$ and is also...
View ArticleSolving for Eigenvalues of a Matrix with a Special Structure (Containing...
I am currently investigating the eigenvalue computation of a matrix with a special structure. Consider an $N$-dimensional matrix where all off-diagonal elements are $a$, diagonal elements are typically...
View ArticleDoes $A:=\{x\in [0,\infty)\,|\,0\leq \{x\},\{x\log x\}\leq 1/2\}$ have...
For a subset $S\subseteq [0,\infty)$, we say that $S$ has positive density if $$\liminf_{T\to\infty}\frac{m(S\cap[0,T])}{T}>0.$$Let $A:=\{x\in [0,\infty)\,|\,0\leq \{x\},\{x\log x\}\leq 1/2\}$,...
View ArticleGeometric Interpretation of the Jacobian Matrix and Its Eigenvectors
I understand that for scalar-valued functions $g: \mathbb{R}^n \to \mathbb{R}$, the gradient represents the direction of maximum ascent. Similarly, for vector-valued functions $f: \mathbb{R}^n \to...
View ArticleTruncation of $C^k$ functions in $C^k$
When we truncate a function $f$ we usually consider $g_n(x)=f(x)\mathbb{1}_{[-n,n]}(x)+f(n)\mathbb{1}_{(n,\infty)}(x)+f(-n)\mathbb{1}_{(-\infty,-n)}(x)$, where $n$ is a natural number. Can we do it...
View ArticleThe relation between Intervals from Dyadic decomposition : No intersection or...
For each $j\in\mathbb Z$, define the collection of half open intervals $D_j:=\{[\ k/2^j, (k+1)/2^j\ )\mid k\in\mathbb Z\}.$I want to show that if $j,\ell\in\mathbb Z$, $I\in D_j$ and $J\in D_\ell$,...
View ArticleDoes $\limsup_{𝑛\to\infty} ( 𝑛*𝑎_𝑛 ) = \infty$ imply that the series $\sum_n...
Consider the series$$ ( \sum a_n ) $$ where $$( a_n \geq 0 ) $$ for all $$( n )$$Assume that$$\limsup_{n \to \infty} (n a_n) = \infty.$$Does this imply that the series $$ ( \sum a_n ) $$ diverges?If...
View ArticleCauchy product for integer sums
For a sequence $(a_i)_{i \in \mathbb{Z}}$ of real numbers we consider the two sided sum: $$\sum_{i=-\infty}^\infty a_i$$We say that this sum is (absolutely) convergent, if both parts:...
View ArticleConstructing the interval $[0, 1)$ via inverse powers of $2$
If $x$ is rational and in the interval ${[0,1)}$, is it always possible to find constants $a_1, a_2, ..., a_n\in\{-1, 0, 1\}$ such that for some integer $n\geq{1}$, $x = a_1\cdot2^{-1} +...
View ArticleEpsilon delta limit definition [closed]
Let $f\colon\Bbb R\to\Bbb R$ be defined as$$f(x)=\begin{cases}\frac1{x^2},&x\neq0\\0,&x=0\end{cases}$$Prove $f$ is continuous at any $c\neq0$ by using $\varepsilon-\delta$ definition of limit...
View ArticleProve $\prod (1+a_i^2) \le \frac {2^n}{n^{2n-2}}\left (\sum...
Problem. Let $n \in \mathbb{N}_{\ge 3}$. Let $a_1, a_2, \cdots, a_n > 0$ such that $\prod_{i=1}^n a_i = 1$. Prove that$$\prod^n_{i=1} (1+a_i^2) \le \frac {2^n}{n^{2n-2}}\left (\sum^n_{i=1}...
View ArticleBounded Quotient of Lipschitz functions should be continuou.
Take $f,g \in C([0,1]) $ Lipschitz, ${{f}\over{g}} \in C((0,1))$ and $\exists \infty >C>0 \forall x \in (0,1) |{{f(x)}\over{g(x)}}| < C $.I want to prove that ${f \over g} \in C([0,1])$ is...
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Number of positive roots of a sum of reciprocals
Is it true that for any $n\geq 1$, any $a_1,\cdots,a_n \in \mathbb{R}_{>0}$ and any $b_1,\cdots, b_n \in \mathbb{R}$, the function:$$f(x) = \frac{b_1}{x+a_1} + \frac{b_2}{x+a_2} + \cdots +...
View ArticleShowing that the set $(0,1]$ is not compact.
Is the following a correct way of showing that the set $(0,1]$ is not compact or I am missing something.We will prove that it does not satisfy the Bolzano Weierstrass Theorem, i.e., there exists a...
View ArticleDetail about an answer of Any open subset of $\mathbb{R}$ is a countable...
I saw an answer to the question that any open subset of Ris a countable union of disjoint open intervals.The link of the beautiful answer is this.I understand the statement in the answer but it remains...
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