Convergence of sequences and different modes of convergence.

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Is the field of real algebraic numbers a complete field?

Let $\mathbb{R}_{alg}$ be the field of real algebraic numbers. Is there exist a metric $|\cdot|$ for which $(\mathbb{R}_{alg}, |\cdot|)$ is a complete field (i.e. any Cauchy sequence converges in ...
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68 views

find $\lim_n\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+…+\frac{a^\frac nn}{n+\frac 1n}$ using Riemann integral

Here is the question: prove that $S_n=\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+...+\frac{a^\frac nn}{n+\frac 1n}$ is convergent for $a>0$ then find its limit. My attempt: If we accept ...
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117 views

Does the harmonic series converge if you throw out the terms containing a $9$?

I found this very amusing comic on the internet the other day: The last frame seems to claim that the harmonic series converges if you throw out all the terms with a $9$ in the denominator. Is this ...
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60 views

Use the comparison test to find whether $\int_0^\infty 1/(x^2+1)^2\,dx$ converges or not

I was thinking what function I should compare it to. If I say whether a function is smaller or bigger than this one, then I must prove that. I was thinking of (x+1)^2 but I realized that this ...
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1answer
44 views

Showing Convergence in Distribution for Conditional Random Variable

I am trying to prove the following: Let $X$ and $Y$ be random variables such that $Y | X = x$ ~ $N(0, x)$ with $X$ ~ $Po(\lambda$). Show that $\frac{Y}{\sqrt{\lambda}} \to N(0,1)$ in distribution as ...
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127 views

Does this series converge, and if so to what value?: $\sum_{n=0}^\infty \left\{\frac{1}{(n+1)^2} \right\}\ln(2n+1)$

I've arrived at this series from a given sequence of terms, but now I'm at a loss as to how to proceed... How does one know which convergence test to use? This isn't a geometric series, so I don't ...
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1answer
28 views

If ${f_n}$ converges to $f$ in $L_p$ sense and to $f'$ point-wisely, does it mean $f=f' a.e.$?

The question came into my mind when I read a theorem from Kubrusly's "Measure Theory: a First Course", saying that if $f_n\rightarrow f'$ uniformly and $f_n\rightarrow f''$ in $L_p$ sense, then ...
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1answer
72 views

Showing Convergence in Distribution of Continuous Function of Sums of R.V.s

I am trying to solve the following: Let $X_1, X_2, . . .$ be i.i.d. r.v.s with mean $\mu$ and positive, finite variance $\sigma^2$, and set $Sn = \sum_{k=1}^{n} X_k, n ≥ 1$. Suppose that $g$ is twice ...
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56 views

Proving a Variation of the the Central Limit Theorem

I am trying to prove the following: Let $X1, X2, . . .$ be positive, i.i.d. r.v.s with mean $\mu$ and finite variance $\sigma^2$, and let $S_n = \sum_{k=1}^{n} X_k$ , $n \ge 1$. Show that $\frac{S_n ...
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1answer
55 views

Convergence of $\int _{-\infty}^{+\infty}\sin(cx)dx$

At this forum there is an abundance of questions regarding the convergence of integrals and sums of infinite series. The mathematicians who answer these questions emphasize that only under strict ...
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1answer
53 views

suppose n is a natural number , prove equation $x^n+nx-1=0$ exist an unique real positive root $x_n$

suppose n is a natural number prove : equation $x^n+nx-1=0$ exist an unique real positive root $x_n$ ; and when $a>1$,$\sum_{n=1}^{\inf}x^a_n$ converges.
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1answer
36 views

Convergence in Probability for a Sequence of Random Variables

I am trying to solve the following: Let $\{X_n, n ≥ 1\}$ be a sequence of i.i.d. random variables with density $f(x) = e^{−(x−a)}$, for $x ≥ a$ and $f(x) =0$, for $x < a$. Set $Y_n = \min(X_1, ...
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25 views

Proving convergence in L1 of a sequence of functions given by integrals

I am required to prove that $x\mapsto\int_x^{x+1/n} n f(y)dy$ converges in the $L^1$ sense to $f$, knowing that $f\in L^1$. My current attempt is: after a variable change, I've rewritten ...
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1answer
59 views

Show that $ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $ as $ \lambda \rightarrow \infty $

Given that the characteristic function for Y is $$ \varphi_Y (t) = e^{\lambda (e^{-t^2/2}-1)} $$ Show that $$ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $$ as $$ \lambda \rightarrow \infty $$ I've ...
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59 views

Convergence in Probability Proof

I am trying to show the following: Let $X_1, X_2, . . .$ be $U(0, 1)$-distributed random variables. Show that $max_{1\leq k\leq n}X_{k} \to 1$ as $n \to \infty$ in probability. I am not sure where ...
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1answer
71 views

Determine convergence of the series $\sum\limits_{n=2}^{\infty}\frac{\cos(\ln(\ln(n))}{\ln(n)}$

How to determine convergence of the series $\sum\limits_{n=2}^{\infty}\frac{\cos(\ln(\ln(n))}{\ln(n)}$ ? I tried to get somewhere with Integral criteria and with comparing to other series but ...
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2answers
45 views

When are both $\sum_{n=0}^\infty \log(a_n)$ and $\sum_{n=0}^\infty a_n$ convergent?

I'm new to this site. Can someone give me some examples of when both: $$\sum_{n=0}^\infty \log(a_n)\qquad \text{ and }\qquad \sum_{n=0}^\infty a_n$$ are convergent?
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1answer
44 views

Can anyone prove D'Alembert Criterion (Dalambert) criterion for converging positive sequences?

This will most likely be on the exam, but it is not given in the text book. In my notebook I have this proof which I will type out, but it makes no sense. Here it goes: $$\text{D'Alembert ...
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178 views

Integrating an infinite sum (statement from paper)

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Define $$v(x,y) = \sum_{k ...
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$u=\sum (u,\varphi_k)_{L^2}\varphi_k$ vs $u(x)=\sum (u,\varphi_k)_{L^2}\varphi_k(x)$ and related questions

I have some doubts after reading several threads. Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of ...
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1answer
34 views

Convergence rate of a function

I'm having a difficult time working out the details of the following problem. I'm hoping someone may be able to point me to a reference or suggest an approach. I have three matrices $(A, B, C)$ and ...
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1answer
27 views

Convergence of subsequence of partial sums implies full convergence?

Define $S_n = \sum_{j=1}^n a_j$ a sum of real numers. Suppose there is a subsequence such that $S_{n_j} \to S$ for some $S$, i.e., the infinite sum converges along this subsequence. Does this imply ...
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62 views

dominated convergence theorem application

Prove or disprove this statement: if $f_n : \mathbb R \to \mathbb R$ are integrable functions with $f_n \to 0$ pointwise and $|f_n(x)| \le \frac{1}{|x| + 1}$ for all $n$, $x$, then $\lim\limits_{n ...
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47 views

Almost sure convergence of $\max(X_1, X_2,\ldots,X_n)$.

$X_1, X_2,\ldots, X_n$ are independent with uniform distribution on $[0,a]$. Prove that $\max(X_1, X_2,\ldots,X_n) \rightarrow a$ almost surely. Ar first I look for the probability distribution i.e. ...
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36 views

Limit of integrals of simple functions over a finite measure

We are given a sequence of simple functions $f_n:\mathbb{R}^2\rightarrow \mathbb{R}$ which converge pointwise to a continuous limiting function $f$. We also have a bunch of positive, finite measures ...
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28 views

How do I find the interval of convergence?

Suppose I have: $$\sum \cfrac{(-1)^n}{\sqrt{n}}x^n$$ If I use the ratio test, I get $$\cfrac{1}{\sqrt{1+\frac{1}{n}}}|x|$$ Why can it be said the radius of convergence here is $1$? Disregard the ...
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17 views

Limits and Convergence of sequences in the form of $(k, k^2, 1/k)$

I'm dealing with proving the convergence and limits of sequences that are defined by multiple points, such as $$ \left(k, k^2, \frac{1}{k}\right) $$ and I'm not sure how to go about doing it. I'm ...
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How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$?

How do I evaluate this sum :$$\displaystyle\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$$ Note : I used many criterions of convergence to show if it converges but i didn't up. Thank you for any ...
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572 views

Integral of odd function doesn't converge?

When I look up $$\int_{-1}^1 \dfrac{1}{x} dx$$ on Wolfram Alpha, it says it doesn't converge. While this is a sum of two diverging integrals, the two areas are clearly symmetric, and I'd assume the ...
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1answer
29 views

Convergence of Series for tangent (only convergence or divergence)

$$\sum_{n=17}^{\infty}\left(\tan\left(\frac{1}{n}\right)\right)^2 \ \ $$ My first guess is to write the series as integral. And use the substitution for u=1/n. That changes my upper and lower ...
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44 views

Prove completeness of a metric space

Let $\mathcal{K} = \{A \subset \mathbb{R}^N| A \neq \emptyset, A \text{ closed and bounded with respect to the euclidean metric} \}$ Let us define $A_\epsilon = \bigcup_{x \in A}U_\epsilon(x)$, where ...
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Find radius of convergence for the given sequence: $\sum_{1}^{\infty} \frac{(-1)^n}{n!}x^n$

I've been trying to realize how to find the radius of convergence for this sequence: $$\sum_{1}^{\infty} \frac{(-1)^n}{n!}x^n$$ I know that it converges for any given $x$, but can someone explain me ...
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62 views

Show that the sequence $\langle b_n\rangle$ Converges to $1$

The following question was asked in my Masters entrance examination but unfortunately I was unable to answer this. Please tell me how to approach this problem correctly. Suppose $\langle ...
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correlated random vectors converge to an independent random vector

Suppose $(X_k,Y_k)_{k\geq 1}$ is a sequence of random vectors. $X_k,Y_k$ have support $[0,1]$ and a joint density function $ f_k(x,y)$ being positive for all $x,y\in [0,1]$ and continuous everywhere. ...
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69 views

Integral test for convergence of $\frac{1}{\ln x}$ [closed]

I want to know if $$\int_0^1 \frac{1}{\ln x}\, dx$$ converges or not.
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3answers
65 views

Determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$

I tried to use D'Alambert theorem to determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$ . $\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} ...
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53 views

Determine convergence of the series $\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$

How to determine convergence of the series: $$\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$$ I spent most of the time using the Integral criteria (since the function $f(x)=\frac{1}{\ln(x)^{\ln(x)}}$ ...
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26 views

Relation between sum of indicators and average of probability for independent sequence

Let $A_1, A_2, \ldots$ be independent events, set $N_n := \sum_{i=1}^n I_{A_i}$ and $\overline p_n := n^{-1} \sum_{i=1}^n P(A_i)$, then $$ P\left( \lim_n \left( n^{-1} N_n - \overline p_n ...
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31 views

Conditionally convergent - limit of series'

Let $\sum_{n=1}^\infty a_n$ be conditionally convergent. Let $k_n:= \max(a_n,0),l_n:=-\min(a_n,0)$ for $n\in \mathbb{N}$ and show that $\sum_{n=1}^\infty k_n =\infty $ and ...
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Describing convergence with probability $1$ in “finite” terms, proof correct

I tried to solve the following exercise: Show that $Z_n \to Z$ with probability $1$ if and only if for every $\varepsilon$ there exists some $n$ such that $P(|Z_k - Z| < \varepsilon, n \le k ...
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15 views

Rearranging series' to converge to a certain point.

Let $\sum_{n=1}^\infty a_n$ a convergent but not absolutely convergent series. 1.1: For $n\in \mathbb{N}$ let $b_n:=max(a_n,0)$, $c_n=-min(a_n,0)$ Show that: $\sum_{n=1}^\infty ...
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14 views

$\nabla\phi_k\stackrel{L^2}{\to}\nabla\phi\Rightarrow\langle\nabla u,\nabla\phi_k\rangle\stackrel{L^2}{\to}\langle\nabla u,\nabla\phi\rangle$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $(\phi_k)_{k\in\mathbb{N}}\subseteq L^2(\Omega)$ with ...
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33 views

Trouble with the proof of convergence of a series.

$\sum_{n=1}^\infty \frac{n}{(-2)^n}$ I tried using D'Alembert's Ratio on it and this is how far I got: $\frac{(n+1)}{(-2)^{n+1}}\frac{(-2)^n}{n}=\frac{n+1}{(-2)\cdot ...
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28 views

Convergence in probability implies boundedness in $L^1$?

Suppose we have a sequence of positive random variables wgich converges to 0 in probability, i.e. $X_n=o_P(1)$. I want to prove that $E[X_n]$ is bounded. My idea: In particular $X_n$ is bounded in ...
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1answer
71 views

The Passare-Tsikh solution to the principal quintic

The Bring-Jerrard quintic, $$x^5+x+t=0$$ can be solved as, $$x = -\sum_{k=0}^\infty(-1)^k\frac{(5k)!}{k!(4k+1)!}\;t^{4k+1}\tag1$$ when, $$|t|<\frac{4}{5^{5/4}}\approx 0.53\dots$$ This paper ...
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4answers
74 views

The sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$

What is the sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$ ? I got that the series converges and the sum seems to be $5$. When trying to explicitly get the sum, I tried to find the ...
0
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1answer
42 views

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then $X_n\to\delta_0$ in distribution

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then prove that $X_n\to\delta_0$ in distribution. Here $\delta_0$ is the degenerate random variable putting all its mass at the point $0$. I ...
0
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1answer
29 views

Snowflake-sequences - Area - Circumference

Consider the following inductively defined snowflakes-sequences: $S_1$ is an equilateral triangle with edge length $l_0$, and $S_{n+1}$ emerges from $S_n$ by dividing each edge by 3 and the middle ...
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39 views

Series' - Convergence - Limit of sequences

Examine the following series' for convergence: a)$\sum_{n=1}^\infty \frac{n^3\cdot 3^n}{n!}$, b)$\sum_{n=1}^\infty\frac{n}{(-2n)^n}$, c)$\sum_{n=1}^\infty \frac{n!}{n^n}$, ...
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3answers
42 views

Convergence of the series $\sum_{n\in\mathbb N}\left(\sin\frac{1}{n^n}\cdot 2^n\cdot n!\right)$ [closed]

Does the series $$ \sum_{n\in\mathbb N}\left(\sin\frac{1}{n^n}\cdot 2^n\cdot n!\right) $$ converge?