How to see this improper integral diverges?
$$\int^\infty_1\frac{1}{x^{1+1/x}}dx$$I'm preparing for exams. I would also like to know what are some commonly used methods to show an improper integral diverges?
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What book is this proof from?
I was searching for a proof for the compactness of [0, 1] and I came across the excellent image of a proof from a book. Can anyone help me identify what book this is?
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What is a contractive mapping vs contraction mapping?
This is an example from a text to show that this mapping does not have a fixed point because it is contractive but not a contraction:I am not sure what the difference is between contractive and...
View ArticleThere isn't $(a_n) \in (-\infty, \infty]^\mathbb{N}$ s.t. $ \forall n\geq 1,...
To solve a measure theory exercise, I assumed the following is true (as it seems very reasonable to me)A neighborhood of infinity (i.e., an interval $(x, \infty]$, where $x \in \mathbb{R}$) cannot be...
View ArticleSuppose that $\lim_{x\to\infty} f '(x) = 0$. Using Mean Value Theorem, show...
Been stuck on this problem for a few days now. It is also given that $f'(x)$ exists for all $x$, and $f'(x)$ can be finite or infinite. My best attempt now is the following, although I'm not 100%...
View ArticleAsymptotics for $f(z) = (z-1) \prod_n \frac{2}{z^{1/a_n}+1}$ with $1/a_1 +...
Let $Re(z) > 1$Consider functions such as$$f(z) = (z-1) ([2 / (z^{1/a_1} + 1)] [2 / (z^{1/a_2} + 1)][2 / (z^{1/a_3} + 1)][2 / (z^{1/a_4} + 1)][2 / (z^{1/a_5} + 1)]...)$$or more formally$$f(z) =...
View ArticleHow to show that if $f^N$ is a contraction mapping, then $f$ has a unique...
If f is a mapping of a complete metric space $(X,d)$ into itself and $f^N$ is a contraction mapping for some positive integer $N$, then $f$ has precisely one fixed point.The Banach fixed point theorem...
View ArticleMeasurable set with respect to sum is also measurable with respect to the...
Let $\mu$ and $\lambda$ be two $\sigma$-finite premeasures on a ring $\mathcal{R}$. The induced outer measure on $\mu$ is$$\mu^* :=\inf\left\{\sum_{n=1}^{\infty} \mu(A_n)\right\}$$where $A_n \in...
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On small remark to Theorem 1.14 in Folland's Real analysis
From here and here we can read about Theorem 1.14 in Folland's Real analysis text and its proof respectively.1.14 Theorem Let $\mathcal A \subset \mathcal{P}(X)$ be an algebra, $\mu_0$ be a premeasure...
View ArticleHelp on formalization of the proof: If $(a,b] =...
If $a<b$ are extended numbers for which $(a,b] = \bigcup_{n=1}^{\infty}(a_n,b_n]$, where $((a_n,b_n])_{n \geq 1}$ is a sequence of disjoint intervals such that $a_n \leq b_n$, so it is clear...
View ArticleProving Proposition 8.2.6 from Terence Tao's Analysis I
I am currently studying Terence Tao's Analysis I and am currently stuck on trying to prove one of the propositions concerning absolutely convergent series over arbitrary sets, which he left as an...
View ArticleQuestion about limits and continuity when the function isn't defined in a...
Let's say I have a function that looks something like this.$$ f(x) = \begin{cases} x & x\leq 0 \\ 1 & x=1 \\ x & x\ge 2 \end{cases}$$What would the values of $$\lim_{x\to1^-} f(x)$$ and...
View ArticleTips on showing collections are $\sigma$-algebras and determining a...
So I am currently enrolled in a measure theory class and as such, a number of homework questions I've come across so far have been along the lines of 'show that some collection is a $\sigma$-algebra on...
View ArticleA Problem On Differentiable Function
Q. Let $f: \mathbb{R}\to\mathbb{R}$ be a differentiable function with $f(0) = 0$. It $\forall x\in \mathbb{R}, 1 \lt f'(x) \lt 2$. Then which of the following statements is true on $(0,\infty)$?(A)....
View ArticleSupremum of a Continuous Function is Continuous
I'm working on this problem from Elementary Analysis by Ross which is intuitive when sketched but keeps stymieing me when I try to write it out.Let $f$ be a continuous function on $[a,b] \subset...
View ArticleUnderstanding some concepts regarding metric spaces.
I already know the base definition and 4 axioms. The things I don't know are completeness, compactness, and connectivity.Also, in the example:$\gamma: [0, 1]\to D$, $\gamma(0) = z_1, \gamma(1) = z_2$...
View ArticleExistence of solutions
Consider a first order equation in $\mathbb{R}$ with $f(t,x)$ defined on $\mathbb{R}\times \mathbb{R}$. Assume the equation $x'=f(t,x)$. Suppose $xf(t,x)<0$ for $|x|>R$ where $R$ is a fixed...
View ArticleReal function resembling a sequence of numbers given by...
A while ago I have asked this question:here about the nature of a certain series which turns out it is divergent in most cases. Now I'm interested about the numbers yielded by nominators of such...
View ArticleSeparate continuity implies measurability
Suppose $f$(x,y) is a function on $\mathbb{R}^{2}$ that is separately continuous: for each fixed variable, $f$ is continuous in the other variable. Prove that $f$ is measurable on...
View ArticleProve that a monotone and surjective function is continuous
Let $I$ be a interval and $f:I \rightarrow \mathbb{R}$ monotone and surjective prove that $f$ is continuous.I tried using the definition of $\epsilon$-$\delta$ and supposing that $f$ is not continuous...
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