Convergence of sequences and different modes of convergence.

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3
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1answer
49 views

Does $\int_0^\infty \frac{1}{1+(x\sin x)^2}\ dx$ converge?

Does the integral $$\int_0^\infty \frac{1}{1+(x\sin x)^2} \ \, \mathrm{d}x$$ converge? I know that I need to look at: $$\sum_{n=0}^\infty \int_{n\pi}^{(n+1)\pi} \frac{1}{1+(x\sin x)^2}\ \, ...
3
votes
1answer
31 views

Proving that the Gamma function $\Gamma(y)$ converges for $y>0$.

How can I justify that $$\Gamma(y)=\int_0^\infty t^{y-1}e^{-t} \, \mathrm{d}t$$ exists for all $y>0$? I'm struggling to compare it to a known convergent integral.
0
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2answers
74 views

Two divergent series such that their product is convergent

I faced a series question it goes something like give an example of 2 divergent series such that when the 2 series are multiplied to each other, the new series becomes convergent, although it looks ...
1
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2answers
35 views

Improper integrals - Showing convergence.

1)Show that for all $n\in\mathbb{N}$ the following is true: $\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$ for a constant $C>0$ and conclude that ...
10
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4answers
383 views

Is there a function that can be subtracted from the sum of reciprocals of primes to make the series convergent

The gamma constant is defined by an equation where the harmonic series is subtracted by the natural logarithm: $$\gamma = \lim_{n \rightarrow \infty }\left(\sum_{k=1}^n \frac{1}{k} - \ln(n)\right)$$ ...
0
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1answer
44 views

Bizarre definition of convergent sequences

I've recently begun studying Analysis, specifically the sequence and series part. I've just come across the definition of convergence: A sequence $(a_n)$ converges to a real number $a$ if, for ...
1
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4answers
87 views

If $x_{n+1}\leq x_n + 1/n^2$ then $x_n$ converges [duplicate]

Let $x_n$ be a sequence of non-negative real numbers such that $\forall n, x_{n+1}\leq x_n+ \frac{1}{n^2}$ Prove that $x_n$ converges. The problem is trivial whenever $x_{n}$ is an ...
1
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1answer
34 views

Determining Probability Generating Function from Probability Mass Function and Convergence

I am trying to solve the following: Suppose $X_{nk}, k=1,2,\ldots,n, n≥ 2$ are i.i.d. random variables $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}\\P(X_{nk}=1)=\frac{1}{n}\\P(X_{nk}=2)=\frac{1}{n^2}$$ ...
1
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2answers
40 views

Struggling with the integrability of $\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$

I know quite a lot tools to determine the integrability of functions, but in this case I really don't know where to start: $$\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$$
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1answer
91 views

Investigate the convergence of $\int _0^\infty \frac{\sin x^2}{x} \ dx$

Investigate the convergence of $$\int_0^\infty \frac{\sin x^2}{x} \, \mathrm{d}x$$ Is it converging? Converging absolutely? I want to use Dirichlet's test for integrals. Let $f(x) = \frac 1 x$ ...
0
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2answers
36 views

How can I solve this integral with the comparison theorem?

I have an integral that I am not sure how to solve with the comparison theorem to see if it is divergent or convergent. $$\int_1^\infty\frac{e^{-2x}}{\sqrt{x+16}}\;dx$$ How can I solve this with ...
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2answers
47 views

Conditional convergence and Riemann's series theorem

There are tests to determine whether an integral or sum is convergent. There are test to determine whether an integral or sum is absolutely convergent. An integral or series is said to be $\mathbf ...
0
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2answers
28 views

suppose $a_n>1$ $a_n$is non-decreasing and bounded. prove: $\sum_{n=-1}^{\infty}(1-\frac{a_n}{a_{n+1}})\frac{1}{\sqrt{a_{n+1}}}$ converges

suppose $a_n>1$, $\{a_n\}$ is non-decreasing and bounded. prove: $\sum_{n=-1}^{\infty}\left(1-\frac{a_n}{a_{n+1}}\right)\frac{1}{\sqrt{a_{n+1}}}$ converges I don't have any idea about how to ...
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3answers
49 views

Question about the Fourier Inversion Formula

We have $$\hat{f}(\xi)=\mathcal{F}f(\xi):= \int_{-\infty}^{\infty}f(x)e^{-2\pi i\xi x}dx,$$ with $f\in L^{1}$, and the Fourier inversion formula says that ...
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0answers
20 views

Almost sure uniform convergence [on hold]

If a sequence of continuous random variables converges almost surely uniform to a random variable, is it true that the limit is $\mathbb{P}$-almos surely continuous?
3
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1answer
71 views

Limit of given expression

Let $\sum a_k=s$. I want to show that $$\lim\limits_{x\to 1^-}(1-x)\sum\limits_{k=1}^{\infty}\frac{ka_kx^k}{1-x^k}=s$$ where $x\in(0,1)$. Thanks for your helps.
0
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2answers
31 views

Limit of Convergent Sequence Property Proof Help

I have a question about this property: Let $\lim\limits_{n\to\infty} a_n = a$, then $\lim\limits_{n\to\infty}(ca_n) = ca$ for all $c \in \mathbb R$ If we consider when $c$ doesnt equal $0$, my ...
0
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0answers
30 views

what does this mean $n\delta$ where $n\rightarrow \infty$ and $\delta \rightarrow 0$

$\lim_{n \rightarrow \infty} sup_{\delta \rightarrow 0} (n \delta)^{-1} |T_n(\theta)_{ij}| < \infty$ a.s, $1 \leq i \leq p$, $1\leq j \leq p$ what does the multiplication of $n$ and $delta$ mean, ...
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3answers
55 views

How can i prove that this integral is convergent/divergent

This is my equation: $$\int_0^{\pi/4} \frac{dx}{x\sin2x}$$ I wish to prove that it's convergent or divergent, by $P$ test and/or comparison test, but it does not seem to be applicable... Is it ...
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3answers
46 views

Finding $\lim\limits_{n\to \infty}({1\over n+1}+{1\over n+2}+…+{1\over n+n})$ using integrals [duplicate]

Finding $\lim\limits_{n\to \infty}\left({1\over n+1}+{1\over n+2}+\dots+{1\over n+n}\right)$. I tried many things but it would work out. I am now studying calculus 2 (In my country the first calculus ...
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1answer
25 views

CDF and Convergence of Maximum of Sequence of i.i.d. R.V. of Random Length

Let $X_1,X_2,...$ be i.i.d random variable $U(0,1)$ distributed. Let $N_m$ be $Poisson(m)$ and independent of each $X_i$. i)Find the cumulative density function of ...
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0answers
18 views

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 ...
3
votes
1answer
70 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 :$$f(x)=\sum_{n=1}^{\infty}\left|\frac{nx^n}{S_{n-1}}-1\right|$$ I need to show that ...
2
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1answer
44 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 ...
0
votes
1answer
29 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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votes
4answers
53 views

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

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 .
0
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3answers
34 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)? ...
6
votes
3answers
82 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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0answers
42 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 ...
1
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2answers
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 ...
7
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2answers
102 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 ...
0
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3answers
59 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 ...
0
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1answer
41 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 ...
3
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4answers
125 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 ...
2
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1answer
26 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 ...
2
votes
1answer
69 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 ...
3
votes
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
52 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
46 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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0answers
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 ...
3
votes
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 ...
0
votes
2answers
54 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 ...
3
votes
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?
5
votes
1answer
39 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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0answers
96 views
+50

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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vote
1answer
24 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 ...
1
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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 ...
1
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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 ...