Convergence of sequences and different modes of convergence.

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limiting distribution of a function of joint normals

Let $Z_n=(X_{1,n},X_{2,n})\sim N(\mu,\Sigma_n)$ where $\mu=(0,0)'$ and $$\Sigma_n=\begin{bmatrix}a^2+\frac1n & ab \\ab & b^2+\frac1n\end{bmatrix}$$ Then where does ...
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25 views

Boundedness of the function

Let $x\in(0,1)$ and $S_{n-1}=\sum\limits_{k=0}^{n-1}x^k$. Then define $f$ as the following :$$f(x)=\sum_{n=1}^{\infty}x^n\left(\frac{n}{S_{n-1}}-1\right)$$ I need to show that ...
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34 views

Prove that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$

Suppose $\{a_i\}_1^{\infty} \subset (0,1)$ a) $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$ b) Given $\beta \in (0,1)$, exhibit a sequence $\{a_i\}$ such that ...
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1answer
17 views

Characteristic Function and Convergence in Distribution of Sequence of R.V.

I am trying to solve the following: Let $X_1,X_2,...$ be a sequence of random variables with $P(X_n=\frac{k}{n})=\frac{1}{n}, k=0,1,2,...,n$. Find the characteristic function of $X_n$ and show that ...
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4answers
51 views

How do I determine $r$ in this geometric series $a+ar+ar^2+\cdots$? [on hold]

The geometric series $a+ar+ar^2+\cdots$ has a sum of $12$, and the terms involving odd powers of $r$ have a sum of 5. What is $r$? Thank you for any help .
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3answers
29 views

Convergence of Sequence Proof Question

I have just learned about the convergence of a sequence with the epsilon definition. So when we try to prove a limit of the sequence, what are we doing essentially (with respect to the definition)? ...
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3answers
75 views

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

How do I evaluate this integral if I supposed that : $a > 0$ $$\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx .$$ For $a=2$ I have got : $2\pi$ I think the result will be : ...
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39 views

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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62 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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92 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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3answers
57 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
39 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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4answers
120 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
24 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
49 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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1answer
46 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
51 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
42 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
34 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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21 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
57 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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2answers
53 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
70 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
43 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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38 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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2answers
51 views

How do I show :$\sum_{n=0}^{\infty }\frac{(-1)^n}{n+1}=\ln2$? [on hold]

How do i show this : $$\sum_{n=0}^{\infty }\frac{(-1)^n}{n+1}=\ln2\text{ ?}$$ Thank you for any help
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59 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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1answer
23 views

$u=\sum (u,\varphi_k)_{L^2}\varphi_k$ 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
32 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
24 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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58 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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43 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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35 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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1answer
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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16 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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1answer
163 views

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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35 views

find the interval of convergence for all x [closed]

$$\sum_{n=1}^\infty \frac{(x+5)^n}{(-3)^n \sqrt{n}}$$ I have been asked to find the interval of convergence. I don't know how go about this problem.
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543 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
24 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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1answer
36 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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53 views

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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58 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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16 views

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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3answers
67 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
61 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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1answer
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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1answer
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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3answers
25 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 ...
2
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0answers
34 views

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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0answers
14 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 ...