Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

0
votes
1answer
21 views

Approximation in $L^2(\Omega)$

I want to prove that if $f_n\to f$ in $L^2(R)$ then $f_n(X)\to f(X)$ in $L^2(\Omega)$ for each random variable X. I think of using the dominated convergence theorem, having the puntual convergence, ...
1
vote
1answer
43 views

A proof on uniform convergence of polynomials with bounded degree

In $[0,1]$ suppose that you have a sequence of polynomials $(P_n)_{n\in\mathbb{N}}$ of at most degree $M$ each. Also, suppose that $P_n(x) \rightarrow 0$ pointwise for every $x\in[0,1]$. Is is true ...
5
votes
1answer
86 views

Proving a strange identity

Numerically, it would seem the following identity holds true: $$\frac{6}{7}=\lim_{n\to\infty}\sqrt[n]{\sum_{k=3}^\infty{\left(k-\sum_{j=1}^{k}\frac{1}{j}\right)^{-n}}}$$ Down below I have proven ...
0
votes
2answers
28 views

How to state convergence through limit comparison test?

I am able to show convergence of the following series through the root test but am trying to practice the limit comparison test and can't figure out how to do it that way. $$\sum_{n=1}^\infty ...
1
vote
1answer
90 views

Is the series: $\,\sum_{n=1}^{\infty}\frac{\mathrm{e}^{-in}}{n}\,$ divergent?

According to mathematica, the complex series $\displaystyle\sum_{n=1}^{\infty}\frac{e^{-in}}{n}$ does not converge. I know that the factor $\dfrac{1}{n}$ in the above series is diverging, but I don't ...
0
votes
1answer
28 views

Conditions of Convergence of a Sequence

$$d_{n}=\sqrt[n]{A^n+B}$$ If $d_{n}$ converges, find an expression for its limit. Explain your conditions of convergence in terms of both numbers and their relationship with each other e.g. "If $A ...
3
votes
3answers
55 views

Approximation of Natural Logarithm using arithmetic.

A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by ...
0
votes
1answer
23 views

True or False, sequences converging.

True or False? For any positive real number r, {r^n} converges. False: Take any positive r, then as n → ∞ r diverges. If {x_n*y_n} converges, then {x_n} and {y_n} both converge. False: Suppose ...
0
votes
1answer
13 views

Proving a limit converges to an equation

Verify that $\lim{\epsilon→0}$ of the given equation converges to the equation of $y=xe^{px}$ $$y=\frac{-1}{\epsilon}e^{px}+\frac{1}{\epsilon}e^{(p+\epsilon)x}$$ After I finish my work, I ended up ...
2
votes
1answer
24 views

Comparing plots question

I have a program that randomly generates line plots and what I would like to do now is compare two of those line plots and get a measure of 'similarity' between them. Now I feel as if this measure of ...
0
votes
1answer
17 views

T/F Limit question.

True or False? a) If $f(x) → 0$ as $x → a^+$,(from the right) and $g(x) \ge 1$ for all $x$ in $\Bbb R$, then $g(x)/f(x) → ∞$ as $x → a^+$. True: take $f(x) = \sin x$ and $g(x) = x^2$ as $x → pi/2$ ...
1
vote
1answer
32 views

True or False, limit, functions questions. Does limit exist?

True or False Let a be a real number, and let f and g be real functions defined at all points x in some open interval containing a except possibly at x = a. a) For each natural n, the function ...
3
votes
1answer
46 views

Why do we care about the 'rapidness' for convergence?

It is those puzzeling improper integrals that I can't get my head around.... Does the (improper) integral $\frac 1{x^2}$ from 1 to $\infty$ coverges because it is converging "fast" or because it has ...
3
votes
1answer
66 views

Convergence of infinite product of prime reciprocals?

Where pn is the nth prime number, does the infinite product $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)$$ converge to a nonzero value? (Any help would be much appreciated!)
1
vote
0answers
17 views

Convergence of a distribution function

I have a problem that has me stumped. Frustratingly enough I can't really get the problem started. It goes as follows: Let $X_{1},X_{2},...$ be independent $C(0,1)$-distributed random variables. ...
0
votes
2answers
63 views

Is the sequence $\{(-1)^n 1/n\}$ divergent?

Is the sequence $\{(-1)^n{1\over n}\}$ divergent or convergent? It's convergent to $0$ because for every $\varepsilon$, exists $1/N < \varepsilon$ by archimedean property, so $|1/n|< ...
0
votes
3answers
37 views

convergente of the sum of sines of the terms of the alternating harmonic series

I want to know about the convergence or divergence of the following series: $$\sum \sin (a_n) $$ where $$a_n=\frac{(-1)^n}{n}$$ The tests that I tried were inconclusive. Is it possible to know? ...
6
votes
4answers
98 views

If $\sum_{n=1}^\infty \frac{1}{a_n}$ converges, must $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ converge?

Suppose $\sum_{n=1}^\infty \frac{1}{a_n} = A$ is summable, with $a_n > 0,$ $n = 1,2,3,\cdots.$ How can we prove that $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ is also summable? This question ...
2
votes
2answers
40 views

Proving a sequence is convergent

Let $\ (x_n )_{n \ge 0} $ be a convergent sequence . Prove that another sequence $\ (y_n )_{n \ge 0} \ $ defined as $ x_n = y_n + 2y_{n + 1} $ is convergent as well . I tried mixing the $\ ...
16
votes
2answers
200 views

Limit of $\int_0^1\frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx$

Set $$u_n= \int_0^1 \frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx\tag1$$ where $\left\{t\right\}=t-\lfloor t \rfloor$ denotes the fractional part of $t$ and where $B_n(\cdot)$ are the ...
0
votes
1answer
33 views

tests for convergence of cos(pi/2n-1)-cos(pi/n)

On my way to saying that the series $\sum_{n=1}^{\infty}(cos(\pi/(2n-1))-cos(\pi/2n))$ diverges, I figured I would re-write the expression $cos(\pi/(2n-1))-cos(\pi/2n)$ as $(-1)^{n+1}cos(\pi/n)$. ...
2
votes
3answers
62 views

Does $\sum_{n=1}^\infty \frac{(-1)^n}{n^{1+\frac{1}{n}}}$ converge?

I want to use the alternating series test here, but I've just been told that it won't work because it's not monotonically decreasing. However, if the alternating harmonic series converges then don't ...
2
votes
1answer
38 views

The interchange of limit for integration

This kind of problem bothers me for a while. Each time I meet such problem I got stuck and has to deal them case by case. So I post this problem here to ask for some general condition of the ...
1
vote
3answers
44 views

Prove {a_n} converges to a?

Let the sequence {b_n} converge to b. Suppose that the sequence {a_n} and the number a have the property that there is a number M and a natural number N such that |a_n - a| ≤ M|b_n - b| for all ...
0
votes
2answers
95 views

Show that $\,a_n=f(1)+f(2)+\cdots+f(n)-\int_1^n f(x)\,dx\,\,$ converges

Let $\,f:[1,\infty)\to \mathbb R\,$ be a decreasing and lower bounded function. Show that the sequence $\{a_{n}\}_{n\in\mathbb N}$ defined as: $$ a_n=f(1)+f(2)+\cdots+f(n)-\!\int_1^n\!\! f(x)\,dx, $$ ...
0
votes
1answer
21 views

Determine whether this integral converges for 0 < p < 1?

$$ \int_{0}^{a} \frac{1}{x^p} $$ I'm lost on how to approach this problem. So far I've integrated it: $$ \int_{0}^{a} \frac{1}{x^p} = \lim_{b\to0}\frac{x^{1-p}}{1-p} \int_{a}^{b} $$ But I don't ...
1
vote
1answer
26 views

Monotone Increasing Function w/ an Upper Bound $\implies$ Unique Solution?

Say I've shown that $f(x)$ is monotonically increasing on $(0, \infty)$ (by showing its derivative is always positive when $x > 0$). Say too I consider $F = \{\alpha\mid f(\alpha) \leq C \}$ so ...
1
vote
0answers
25 views

Convergence of series, cesaro summability

Suppose that series $\sum_{n=1}^\infty na_n^2<\infty$ and $\sum_{n=1}^\infty a_n$ is Cesaro summable. How do you show that $\sum_{n=1}^\infty a_n$ converges? I know that if $\lim_{n\rightarrow ...
0
votes
0answers
17 views

Meaning of $\liminf \frac{an}{bn} > 0$ and $\limsup \frac{an}{bn} < \infty$

I am having trouble understanding what is meant by $\liminf_{N}\frac{a_{N}}{b_{N}} > 0$ and $\limsup_{N}\frac{a_{N}}{b_{N}} < \infty$ for $N\to \infty$. I think that it has something to do with ...
1
vote
0answers
28 views

Does the series $\sum_{n=1}^\infty \frac{(-1)^n}{n^{1+\frac{1}{n}}}$ converge?

Attempt: We can write the terms in the series as $(-1)^n a_n$ where $$ a_n = \frac{1}{n^{1+\frac{1}{n}}}< \frac{1}{n}.$$ And since $\lim_{n \to \infty} \frac{1}{n} = 0$ and is monotonically ...
2
votes
0answers
22 views

Fourier transform of an exponentially singular radial function

I am trying to compute the 3D Fourier transform of a spherically symmetric function of the form $$f(r) = e^{\frac{1}{r} e^{-r}} - 1\, ,$$ which entails the integral $$\begin{aligned}F(k) =& \int ...
2
votes
2answers
54 views

Prove sequence $S_n$ converges

If $S_1 = \sqrt{2}$, and $S_{n+1} = \sqrt{2 + \sqrt{S_n}}$ (n = 1,2,3....), prove that $\{S_n\}$ converges, and that $S_n < 2$ for all $n \in \Bbb{N}$ This is one the questions ...
1
vote
5answers
77 views

Does the series $\sum_{n=1}^\infty \frac{n+1}{n^3+10n}$ converge?

Using the ratio test, we evaluate: $$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty}\left| \frac{(n+1) + 1}{(n+1)^3 + 10(n+1)} \cdot \frac{n^3+10n}{n+1} \right| = \lim_{n ...
0
votes
3answers
41 views

Prove sequence $\frac{5n+1}{n^{5}-2}$ is convergent

Please can someone help prove that the sequence $\frac{5n+1}{n^{5}-2}$ is convergent from first principles? Thanks
1
vote
2answers
32 views

sequence ${1/n}$ converges in R (to 0) but fails to converge in the set of all positive real number. Why?

Sequence $\{\frac{1}n\}$ converges in $\mathbb{R}$ (to $0$) but fails to converge in the set of all positive real numbers. I don't fully understand why it fails to converge in the set of all positive ...
0
votes
2answers
22 views

Convergence of $\sum_{n = 0}^{\infty} \frac{qz^n}{1 - qz^n} $

I need to prove that the following series converges $$\sum_{n = 0}^{\infty} \frac{qz^n}{1 - qz^n}$$ for $|q| < 1$, $|z| < 1$. I thought maybe to use some comparison tests, but I'm not sure if ...
0
votes
2answers
30 views

Real Analysis: Limit Proof with help of ratio convergence property

The thing is to, using the property above, prove the expression below. First tried to obtain the difference between n and n+1 terms, then factored out n+1 ^ p and rearanged the sequence. I've ...
2
votes
3answers
54 views

Absolute convergence of $\sum_{n=1}^{\infty} (-1)^n (1-\cos1/n)$

I would like to see if $$\sum_{n=1}^{\infty} (-1)^n \left(1-\cos \frac{1}{n}\right)$$ converges absolutely.
3
votes
4answers
51 views

$\sum_{n=1}^\infty \frac{n+1}{\sqrt{n^3+1}}$convergent/divergent?

Please could someone help prove $$\sum_{n=1}^\infty \frac{n+1}{\sqrt{n^3+1}}$$ converges/diverges? Thank you.
0
votes
2answers
17 views

Convergence of this alternating series?

I "heard" the following formula for any $C \ge 1$: $\sum\limits_{k=0}^\infty \dfrac{(-1)^k}{(k+1)C^k} = C \log \dfrac{C+1}{C}$ Is it correct? What would be a proof?
-1
votes
1answer
41 views

If limn→∞(an − bn) = 0 and sum to infinity (bn) converges, then sum to infinity (an) converges. True or false? [closed]

If $\lim_{n\to\infty}(a_n − b_n) = 0$ and $\sum b_n$ converges, then $\sum a_n$ converges. I need to prove whether this is true or false, and if it is false can anyone please give a counter example? ...
5
votes
3answers
136 views

How to calculate convergence of this function?

Given $n$, is there any easy way to calculate convergence of this summation. $$\sum_{k=0}^\infty\dfrac{1}{^{n+k}C_n}$$ EDIT: Also I need to find at which value this series converges.
3
votes
1answer
65 views

Evaluating a trigonometric product $\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$

I'm interested in finding a closed form for $$\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$$ Wolfram Alpha confirms that it converges, but I can't find any plausible closed forms. I've made ...
2
votes
3answers
72 views

Prove that the distance between 2 Cauchy sequences is convergent.

Here is the exact question: Let $(S,d)$ be a metric space. Let $(p_n)$ and $(q_n)$ be two Cauchy sequences in $(S,d)$(note that these two sequences are not necessarily convergent since $(S,d)$ is ...
1
vote
0answers
22 views

Practical Intuition behind converge of random sequence $X_n \to X$

Can someone come up with a practical example as to what it means for a set of random variables to converge i.e. $X_n(\zeta) \to X(\zeta)$? Specifically, what is the meaning of $n$ and $\zeta$?
4
votes
1answer
92 views

Why does my money converge to zero?

I invest $\$1000$ in a stock. Every year, the stock price will either increase by $90\%$ or decrease by $50\%$. The expected value of the change in the price is: $0.5\cdot90\% + 0.5\cdot(-50\%)= 20\% ...
0
votes
0answers
26 views

Convergence of random variables depending on the measure

Suppose the probability spaces $\left([0,1], \mathcal{B}([0,1],\mu_i \right)$ for $i=1,2,3$ , where $$ \mu_1 = \lambda , \ \ \ \ \ \ \ \ \ \ \mu_2 = \delta_1,\ \ \ \ \ \ \ \ \ \ \mu_3 = ...
1
vote
1answer
36 views

Check the convergence (& absolutely) of parametric integral [closed]

$$\int\limits_{-1}^{1} \left(\frac{1+x}{1-x}\right)^{\alpha} ln(2+x)dx$$ Don't know where to start..
3
votes
2answers
98 views

How to show that $\sin(n)$ does not converge?

I am supposed to show that $\sin(n)$ does not converge by constructing two subsequences: one subsequence contains terms of $\sin(n)$ that are between $1/2$ and $1$, and the other subsequence contains ...
1
vote
1answer
31 views

proof of DCT with weak condition(almost everywhere)

I have a question about a proof of the dominating convergence theorem, with weak requirements. Before I show the proof from the book, note that in my book you are allowed to integrate functions that ...