5 votes

How to solve this mathematical analysis limit problem?

Let $a_n$ denote $\sum_{k=1}^{n}\left(\dfrac{1}{n}+\dfrac{k}{n^2}\right)^{1+\frac{2k}{n^2}}$ Since we know $\frac{1}{n}+\frac{k}{n^2}<1$, Then: $a_n \leq$ $\sum_{k=1}^{n}(\frac{1}{n}+\frac{k}{n^2})...
Scott Hahn's user avatar
  • 1,142
5 votes
Accepted

Given continuous $f:\mathbb{R}\to\mathbb{R}$ and compact $K\subseteq f(\mathbb{R})$, show there exists a compact set $C$ such that $f(C)=K$.

Let $a=\inf K$ and $b=\sup K$. By Let $f : I \to \mathbb R$ be continuous. For any compact interval $J \subseteq f(I), \exists$ a compact interval $K \subseteq I$ with $f(K)=J.$ we see that there is ...
geetha290krm's user avatar
  • 24.5k
4 votes
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Proving $\lim_{n \to \infty}\int_n^{n+1}\frac{\sqrt{x-1}}{\sqrt{x^2+1}}dx=0$

We have that $$0\le \int_n^{n+1}\frac{\sqrt{x-1}}{\sqrt{x^2+1}}dx \le \int_n^{n+1}\frac1{\sqrt{x}}dx=2(\sqrt{n+1}-\sqrt n)=\frac2{\sqrt{n+1}+\sqrt n}\to 0 $$ As noticed in the comments, more simply $$...
user's user avatar
  • 148k
4 votes
Accepted

Is $\mathbb{R}\cup\{-\infty,+\infty\}$ the categorical co-completion of $\mathbb{Q}$

The extension $L$ you describe does not preserve colimits in general. For instance, let $\mathcal{E}$ be the poset $\{0,1\}$ and let $D$ send the negative rationals to $0$ and the nonnegative ...
Eric Wofsey's user avatar
4 votes
Accepted

Why does $\lim_{x\to\infty}\prod\limits^{\lfloor x\rfloor}_{n=1}\frac{\lceil x\rceil}{\lfloor x\rfloor}=e$?

Let $k=\lfloor x\rfloor$ $$\prod\limits^{\lfloor x\rfloor}_{n=1}\frac{\lceil x\rceil}{\lfloor x\rfloor}=\prod\limits^{k}_{n=1}\frac{k+1}{k}=\left( 1+\frac1k\right)^k$$ therefore, $$\lim_{x\to\infty}\...
MathFail's user avatar
  • 18.4k
3 votes
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Find the function $f(x)=\sum_{n=1}^{\infty}3(-1)^{n+1}(3x)^{n-1}$

$$f(x)=\sum_{n=1}^{\infty}3(-1)^{n+1}(3x)^{n-1}$$ Let $m=n-1,$ $$f(x)=\sum_{m=0}^{\infty}3(-1)^{m+2}(3x)^{m}=3\sum_{m=0}^{\infty}(-3x)^{m}=\frac{3}{1+3x}$$
MathFail's user avatar
  • 18.4k
3 votes

Proving $\lim_{n \to \infty}\int_n^{n+1}\frac{\sqrt{x-1}}{\sqrt{x^2+1}}dx=0$

Define $$F(x) := \int_1^x \frac{\sqrt{x-1}}{\sqrt{x^2+1}}\, dx$$ By the Fundamental Theorem of Calculus, $F$ is differentiable, and $$F'(x) = \frac{\sqrt{x-1}}{\sqrt{x^2+1}}$$ for all $x > 1$. ...
stoic-santiago's user avatar
2 votes

Compute the integral $\int_0^\infty t^{3/\xi-1} e^{-t} \Gamma(\frac{2}{\xi},t) \ dt$

The only closed form I see is via the incomplete beta function: $$\int_0^\infty t^{\alpha-1}e^{-t}\,\Gamma(\beta,t)\,dt=\Gamma(\alpha+\beta)\,\mathrm{B}(1/2;\alpha,\beta),$$ obtained using the ...
metamorphy's user avatar
  • 37.4k
2 votes
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sequence of non-negative functions $f_n$ tending to $0$ pointwise with $\int f_n \to 0$, but no integrable $g$ such that $f_n \leq g$ for all $n$.

One way of coming up with an example is to let $f_n$ be a triangle function supported on $[n,n+1]$ with height $\frac{2}{n}$, so that $\int f_n=\frac{1}{n}$. Then, $f_n\to 0$ pointwise and $\int f_n\...
peek-a-boo's user avatar
2 votes

Is "uniform continuity theorem" provable without using sequences?

You can, in fact, prove uniform continuity in this style… with a slight modification. (The flaw in Cactus’s counter-argument – though their proof is the “better” one – is that even for a function $\...
Aphelli's user avatar
  • 31.7k
2 votes

The set $X := [1, 2] \cup [3, 4]$ is not connected

When we are dealing with metric subspaces like $X$, the notion of something being "relatively open" is exactly equivalent to openness in the subspace. That is, we have that for instance $[1,...
kodiak's user avatar
  • 283
2 votes

Is $\mathbb{R}\cup\{-\infty,+\infty\}$ the categorical co-completion of $\mathbb{Q}$

The free cocompletion of any category is its category of presheaves. The category of presheaves on $\mathbb Q$ is the category of families of sets, contravariantly indexed by rational numbers. This is ...
Kevin Arlin's user avatar
  • 50.8k
1 vote

Is "uniform continuity theorem" provable without using sequences?

You can't do it like that. Indeed, if such an argument worked, you could use the same reasoning to prove that any function $\mathbb{R} \rightarrow \mathbb{R}$ is uniformly continuous on $\mathbb{R}$ (...
Cactus's user avatar
  • 1,732
1 vote

Decide whether $f$ is differentiable in $(0,0)$ or not given its directional derivative

If $f$ is differentiable in $a$ then you know that for $v = (v_1,v_2)$ you should have $$ \frac{\partial f}{\partial v} (a)= (\nabla f (a)) \cdot v = v_1 \frac{\partial f}{\partial e_1}(a) + v_2 \...
Digitallis's user avatar
  • 2,796
1 vote
Accepted

Prove $\exp(\int^x_0 \frac{U(t)}{t} dt)$ is reguarly varying (Extreme Values, Regular Variation and Point Processes)

Let's denote $f(x) = \exp \left(\int\limits_0^x \frac{U(t)}{t} {\rm d}t\right)$. Following the given theorem, one needs to prove that $\int\limits_0^1 \ln \left(\frac{f(x)}{f(\lambda x)}\right) {\rm d}...
Yalikesifulei's user avatar
1 vote

The set $X := [1, 2] \cup [3, 4]$ is not connected

Openness and closedness depends on the ambient space. In the very common ambient space of the real numbers, $[1,2]$ is a non-open closed subset, as there does not exist any $r > 0$ such that we ...
Lemmon's user avatar
  • 1,034
1 vote

The set $X := [1, 2] \cup [3, 4]$ is not connected

I think you can agree that in any metric space $(X,d)$, the open balls $B_r(x)$ with radius $r$ centered at $x$ are open. In fact, by definition a subset $Y\subseteq X$ of a metric space is open if ...
Vercassivelaunos's user avatar
1 vote

Can this proof of the second MVT for integrals be salvaged?

It seems unlikely that we can extend this argument to cases where $f$ changes sign on $[a,b]$. We can easily construct examples where $\int_a^b fg$ does not lie between $A$ and $B$. Let $g(x)=x$, $f(x)...
Kevin.S's user avatar
  • 3,159
1 vote
Accepted

A question about the d'Alembert ratio test for convergence.

Suppose we do have some sequence $\{ a_n \}$ that fulfills the property as described, i.e. $\lim_{n \rightarrow \infty}|\frac{a_{n + 1}}{a_n}| = L > 1$. With this assumption, we can choose some $\...
kodiak's user avatar
  • 283
1 vote

Local minima and derivative

transfer everything except for o(h) on the LHS, use the definition of o(h) - particularly the definition of a limit (epsilon delta definition), put epsilon := (1/2)|f′(x∗)|. Normal people use epsilon ...
Jan Kubica's user avatar
1 vote
Accepted

Can the convergence of inner products be 'closed'?

What follows is an example that the uniform convergence on $K$ does not suffice. Let $\{e_j\}_{j=1}^\infty$ be an orthonormal basis. Consider $$v_n=\sum_{j=1}^n 2^{-j}e_j,\quad v=\sum_{j=1}^\infty 2^{-...
Ryszard Szwarc's user avatar

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