0 or infinite limit in limit comparison test for improper integral
In this doc, the steps for using the Limit Comparison test are:Use the LCT when trying to determine whether $\int_{a}^{\infty} f(x) \, dx$ converges and the function $f(x)$ is positive and looks...
View ArticleFind lim sup and lim inf of $((-1)^n+1)+1/(2^n)$
Here is what I have, but I do not know if my understanding of lim sup and lim inf are correct:lim sup= 2because the smallest sup of the sequence would be $(1+1)+0=2$lim inf=0because the largest inf of...
View ArticleFor a family of sets {$A_n $}$_{n\in\mathbb{N}}$ given by $A_n=(0,n)$ prove...
The set of subsets of a partricular set $S$ defined as {$A_i: i \in I$} where $\forall i \in I, A_i$ is a unique subset of $S$, is called the family of subsets of $S$ indexed by $I$. In this case...
View ArticleAttempted Proof of a Theorem on Nested Intervals
I am trying to prove the Nested Interval Theorem, which is:Given nested closed intervals of Real Numbers $$[a_1,b_1]\supset[a_2,b_2]\supset\cdots\supset[a_n,b_n]\supset\cdots$$ and...
View Article$X$ is measurable w.r.t $\sigma(X_1,X_2)$, then $X=f(X_1,X_2)$ for some...
It is relatively easy to show that $X$ must be a function of $X_1,X_2$. One just need to verify that $X$ must be constant on any level set $\{\omega|(X_1(\omega),X_2(\omega))=(a,b)\}$.It would be nice...
View ArticleHow to prove these properties from the set Σ_m of null functions?
QuestionA function $f : \mathbb{N} \to$ {0, 1} is called finally null if there exists $k \in \mathbb{N}$ such that, for all $n \geq k$, $f(n) = 0$. In this case, the rank of $f$, denoted by...
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Unexplained conclusion regarding an inequality
How come does the author deduce that from:$$\frac{\epsilon}{2^i}\sum\limits_{k \in T}\Delta x_k < \frac{\epsilon}{4^i}$$We would get:$$\sum\limits_{k \in T}\Delta x_k < \frac{\epsilon}{2^i}$$It...
View ArticleLimit points of $\left\{\frac{1}{2^m}+\frac{1}{3^n}+\frac{1}{5^p}\ : \...
I intend to find the limit points of the set $\mathcal S = \left\{\frac{1}{2^m}+\frac{1}{3^n}+\frac{1}{5^p}\ : \ m,n,p\in\mathbb N\right\}$.These are the ones I could think of:$S_1 =...
View ArticleLee, introduction to smooth Manifolds, problem 1-9
I am taking this class that follows John Lee's introduction to manifolds, but I have not yet taken a complex analysis class (I have just been pushing it to the last semester of my undergraduate). I am...
View ArticleLower bound for $A_2$ functions
Suppose that we have a function $f$ that is of exponential type, and that the function $|f(x-i)|^2 \in A_2$ is such that it satisfies $$\left(\frac{1}{|B|} \int_B |f(x-i)|^2\right)\left(\frac{1}{|B|}...
View ArticleUniform Convergence and ODE
The following problem appeared on a past exam at my institution:Suppose that for each $n\in\mathbb{N}$, $u_n:\mathbb{R}\rightarrow\mathbb{R}$ is a differentiable function satisfying...
View ArticleBound an integral from below
the following question appeared in my calc exam:prove that there exists some $c>0$ such that for all $n \in \Bbb N$,$$cn \le\int_0^n f(x)dx$$whereas $f(x)=x^2-floor(x^2)$I've tried evaluating it...
View ArticleRiemann Integrable
Let $f: [a,b] \rightarrow \mathbb{R}$ and $g: [a,b] \rightarrow \mathbb{R}$ be bounded. Suppose $S = \{x \in [a,b] : f(x) \neq g(x)\}$ is finite. Prove that if $f$ is Riemann integrable on $[a,b]$ then...
View ArticleWhat sorts of operators satisfy this condition?
I'm trying to find some operators $P:\mathbb{X} \to \mathbb{R}^n$ that satisfy these properties (it is okay to assume $\mathbb{X} \subset \mathbb{R}^n$ is compact):(i) $\|\sum_{k=1}^N p_{i} P(x_i) - x...
View ArticleThe differential is NOT the Jacobi Matrix?
In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition:Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to...
View ArticleAnalyzing the asymptotic behavior of a given function $f(\mu, t)$ as $t \to...
Consider the following improper integral$$f(\mu, t) = \frac{2}{\pi} \int_0^\infty \frac{\sin u}{u} \frac{\left(U+u\right) \exp\left(-\frac{\mu}{t} \, u\right) - 2U \exp\left(-\frac{\mu}{t} \, U...
View ArticleProving $r(t) \to 0$ for a spiral
This is a problem from Ted Shifrin's book on Differential Geometry; consider the spiral $\vec{\alpha} (t) = {r}(t)(\cos t, \sin t)$ where $r(t)$ is a $C^1$ function bounded between $[0,1]$....
View ArticleIf $\sum a_n$ and $\sum \ln(1+a_n)$ converge, does $\sum a_n^2$ converge?
Let $(a_n)_{n=1}^\infty$ be a sequence with the property $a_n>-1$ for all $n$. Assume $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty \ln(1+a_n)$ converge. Must $\sum_{n=1}^\infty a_n^2$ converge...
View Article$\lim_{x\to 0^+}\sum_{n=1}^\infty \frac{\chi(n)}{n} e^{-nx}$
In my answer to a question on this site I googled for a reference and found this. The answers of the user reuns of the latter linked question goes to great length to show that$$\lim_{x\to...
View ArticleFind a constant $c > 0$ such that for all $n \in \mathbb{N}$, the following...
The problem is to find a constant $c > 0$ that holds for all $n \in \mathbb{N}$.Now, what I have tried is the following:I wrote $\{x^2\} = x^2 - \lfloor x^2 \rfloor$ and split the integral into two...
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