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Questions tagged [convergence]

Convergence of sequences and different modes of convergence.

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1answer
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Understanding the solution key to a problem which shows that the integral of a sum equals a given value.

Suppose that the domain of convergence of the power series $\sum_{k=0}^{\infty} c_{k}x^{k}$ contains the interval $(-r, r)$. Define $$f(x) = \sum_{k=0}^{\infty} c_{k}x^{k} \hspace{1cm} \text{ &...
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4answers
34 views

Prove that $\sum_{n=1}^∞\frac{\left(\ln n\right)^3}{n^3}$ is a convergent series by using comparison test

I proved by using the integral test that the series is convergent but can't find a way to prove by using the comparison test, which was required.
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1answer
25 views

Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove the following properties.

Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove: 1) If $\alpha > 0$, show $\sum_{n = 0}^{\infty} a_{n}x^{n}$ converges if $|x| < 1/\alpha$ and diverges if $|x| > 1/\alpha$ ...
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0answers
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For each interval $[a,b]$ contained in $I$, sequence $\{f_{n}:[a,b]\rightarrow\mathbb{R}\}$ converges uniformly to $f : [a,b] \rightarrow\mathbb{R}$

$I$ is an open interval Using the following fact to show this: ${\{f_n\}}$ converges pointwise on $I$ to the function $f$, and ${\{f'_n}\}$ converges uniformly on $I$ to the function $g$ Attempt: ...
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1answer
32 views

Proving $\sum_{k = 0}^{\infty} \frac{1}{1 + |x|^{k}}$ converges if and only if $|x| > 1$

I would like to show $$\sum_{k = 0}^{\infty}\frac{1}{1 + |x|^{k}} $$ converges if and only if $|x| > 1$. I think that the best way to show the backwards direction is to assume we havve $|x| \leq ...
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1answer
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Proving the sequence $f_{n} = \sqrt{x^{2} + 1/n}$ converges uniformly to $f(x) = |x|$ on $(-1, 1)$.

I have the following exercise from my book: For each $n \in \mathbb{N}$ and each $x\in (-1, 1),$ define $$f_{n}(x) = \sqrt{x^{2} + \frac{1}{n}}$$ and define $f(x) = |x|$. Prove that the ...
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0answers
28 views

Convergence of a series involving cosine

Let $x \in (0, 2\pi)$. Is the series $\sum_{n=1}^{\infty} \frac{\cos(n^2x)}{n}$ convergent? My guess is: YES and I would like to use Dirichlet test: however I have troubles proving that the partial ...
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1answer
5 views

Find the “region of interest” of an unknown function

Given an unknown function $f:\mathbb{R} \rightarrow \mathbb{R}$, is it possible to find it's region of interest? By that I mean either the range in which $f$ does not converge or diverge, e.g. $f(x)=(...
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1answer
22 views

Formulated a Series Problem But Unfortunately Don't Know How To Solve

This all started when I was playing around with a financial spreadsheet. There is no need to know financial terms as I've managed to convert this observation into a mathematical problem. But just in ...
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2answers
31 views

How this series a_j converges?

We have $a_n\geq 0$ and suppose that $\sum_{j=n}^{2n} a_j\leq \frac{1}{\sqrt{n}}$. I dont know how to derive that $\sum a_j$ converges.
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2answers
49 views

Does $\int_1^\infty \frac{\ln(x)}{x^2} dx $ converge or diverge?

I tried to solve it in an intuitive manner, but I am not sure if it's right or wrong. Some feedback would be lovely! This is how I approached the problem. Step 1: I used integration by parts. $ \...
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0answers
38 views

Convergence of $b_n = \frac{\sum_{i=1}^na_i}{4}$

Question: Let $\{a_n\}_{n\in \mathbb N}$ be a sequence of real numbers, and for each $n\in \mathbb N$ define $$b_n= \frac{\sum_{i=1}^na_i}{4}.$$ Prove that if $\{a_n\}$ converges to $A$, then so ...
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0answers
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Trying to Prove a generalization of Raabe's test

I'm trying to prove (or disprove) a statement by Feld: https://math.feld.cvut.cz/mt/txte/2/txe3ea2d.htm It's a generalization of Raabe's test and it says that for a sequence $a_n$, for all n, if $...
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0answers
17 views

Is the (x-a) format necessary when finding the radius of convergence for a geometric series?

So I am taking AP Calculus BC, and we are currently working on convergence and divergence of series. I came across the following problem in one of my homework assignments: Here is the work I did to ...
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0answers
33 views

$\lim_{n\to\infty} n C_n =0$ where $\sum^\infty\lvert C_n\rvert$ is convergent? [duplicate]

I am trying to prove that if $\sum_{n=1}^\infty\lvert C_n\rvert$ is convergent, then $\lim_{n\to\infty} n C_n =0$. It seems like it should be simple, but I can't figure it out.
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0answers
23 views

If $f(t,x)$ is continuous and $B_{t}$ has continuous paths, then $f(t,B_{t})$ converges almost surely

Let $f \colon [0,\infty) \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a function which is continuous in both variables $t$ and $x$. Let $(B_{t})_{t \in [0,\infty)}$ be a stochastic process ...
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3answers
87 views

The series $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+…$

Consider the expression $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+...$ Denote the numerator and the denominator of the $j^\text{th}$ term by $N_{j}$ and $D_{j}$, respectively. Then, $N_1=1$, ...
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3answers
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find a free ultrafilter on $\Bbb N$

Suppose $(x_n)_n$ is a bounded sequence of complex numbers, there must exist a accumulation point, say $x_0$, thus we can find a free ultrafilter $\mathcal{F}$ on $\Bbb N$ such that $\lim_{\mathcal{F}}...
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2answers
36 views

Questions about conditionally convergent series and rearrangement of [on hold]

According to Riemann Series Theorem or Riemann Rearrangement Theorem a conditionally convergent series - with a clever rearrangement of terms - can converge to any desired value, or even can be shown ...
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0answers
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Probability - scaled convergence in probability implies scaled almost sure convergence for an increasing sequence of RVs

I'm studying for a probability exam, and was having trouble with this problem: Let $X_n$ be an increasing sequence of random variables. Prove or disprove - If $X_n/n\rightarrow 1$ in probability, ...
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3answers
75 views

What value does $\sum_{n=1}^{\infty} \dfrac{1}{4n^2+16n+7}$ converge to?

What value does $$\sum_{n=1}^{\infty} \dfrac{1}{4n^2+16n+7}$$ converge to? Ok so I've tried changing the sum to: $$\sum_{n=1}^{\infty} \dfrac{1}{6(2n+1)}-\dfrac{1}{6(2n+7)}$$ and then writting ...
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1answer
25 views

Prove: $\sum b_n < \infty \Longrightarrow \sum a_n < \infty \ $, where $ \ \exists N: \forall n \geq N: \frac{a_{n+1}}{a_n} \leq \frac{b_{n+1}}{b_n}$

Given two positive sums: $\sum_{n=1}^{\infty} a_n \ $ and $ \ \sum_{n=1}^{\infty} b_n $. $ \ \ \exists N: \forall n \geq N: \ \frac{a_{n+1}}{a_n} \leq \frac{b_{n+1}}{b_n}$ Prove: $\sum {b_n} < ...
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5answers
73 views

Convergence or divergence of $ \sum_{n=1}^{\infty}\frac{\sqrt{n+2}-\sqrt{n}}{n^{3/2}} $: How to argue?

Does the series $$ \sum_{n=1}^{\infty}\frac{\sqrt{n+2}-\sqrt{n}}{n^{3/2}} $$ converge or diverge? My attempt was to write the series as $$ \sum_{n=1}^{\infty}\frac{\sqrt{n+2}-\sqrt{n}}{n^{3/2}}=\...
4
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1answer
64 views

Real Analysis, $\lim\limits_{n\rightarrow\infty} \int_{0}^{1} \frac{e^{-nt}-(1-t)^n}{t} dt$

I was trying to compute $\lim\limits_{n\rightarrow\infty} \int_{0}^{1} \frac{e^{-nt}-(1-t)^n}{t} dt$ using Lebesgue's dominated convergence THM, but I can't exactly figure out how to do. I mean, I ...
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0answers
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Can Convergence in probability in this problem be reinforced to Almost sure convergence

$X_1,X_2,…,X_n$ are independently and identically distributed and $E(X_i)$ exists, $\mu_n=E(X_n I(X_n \le n)),S_n=\sum_{i=1}^n X_i$. Proof:$$\frac{S_n}{n}-\mu_n\overset{p}{\to }0$$ My answer is: $$\...
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3answers
49 views

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

I have to find out if the series $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{3n+n(-1)^n}$$ converges. Root test and ratio test did not work out for me. I also tried the alternating series test, but I can ...
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1answer
52 views

$\frac {2}{5} + \frac {4}{45} + \frac {8}{765} + \cdots =?$ [on hold]

How do I find the infinite sum? $\displaystyle\frac{2}{{{2}^{2}+{1}}}+\frac{{{2}^{2}}}{{{\left({2}^{2}+{1}\right)}{\left({2}^{3}+{1}\right)}}}+\frac{{{2}^{3}}}{{{\left({2}^{2}+{1}\right)}{\left({2}^{...
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0answers
51 views

How to prove that $\sum_{n=1}^\infty a_n$ is convergent

Given that $|a_{n+1}-a_{n}| < 1/n^2$ and $\lim \limits_{n \to \infty} a_n = 0$, I have to prove that $\sum_{n=1}^\infty a_n$ is convergent. I've found no proof yet. Please help me.
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1answer
21 views

Problem on Weak Law of Large Numbers

Question- $X_n$ can take only two values $n^a$ and $-n^a$ with equal probabilities. Show that we can apply weak law of large numbers to the sequence of independent random vatiables ${X_n}$ if $a<\...
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2answers
62 views

Why doesn't $\sum \frac{sin(\frac{1}{n})}{\sqrt(n)}$ diverge?

Why doesn't $\sum \frac{sin(\frac{1}{n})}{\sqrt(n)}$ diverge? I know that it converges, I just want to know what i'm doing wrong here. Here's what I did. $\sum\frac{-1}{\sqrt n} < $ $\sum \...
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0answers
37 views

Does this statement related to the $L^2$ convergence hold? [on hold]

I need to prove that a random variable sequence $X_n$ is a Cauchy sequence. Reading some notes I found the following statement: Assuming that $X_n$ is a random sequence with $E(X_n^2)<\infty$ $\...
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0answers
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Proof of a convergence of sets in the context of Finite Perimeter sets

Let $E \subset \mathbb{R}^n$ be a set of finite perimeter that satisfies $ \mathcal{L}^n (E) < \infty$. Assume that $E$ is symmetric with respect to the hyperplane $\{x_n = 0\}$. We know that there ...
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0answers
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Let $X_1, X2,\dots, X_n$ be iid variables with cdf $F(x)$. Define the empirical cdf as

Let $X_1, X_2,\dots, X_n$ be iid random variables with CDF $F(x)$. Define the empirical CDF as $$ F_n(x) ={1\over n} \sum_{i=1}^n I(X_i \le x)\quad -\infty < x < +\infty $$ where $I(.)$ is ...
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1answer
36 views

test the integral $\int_{0}^{\infty} \frac {x}{3x^4 + 5x^2 +1}dx$ for convergence

test the integral $\int_{0}^{\infty} \frac {x}{3x^4 + 5x^2 +1}dx$ for convergence. My thought Can I compare it with 1/(3x^4)? Any hints for the solution are appreciated!
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0answers
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Convergence of gradient descent method for functions without Lipschitz gradient

In Bertsekas' book "Parallel and Distributed Computation: Numerical Methods", it is stated in Exercie 2.1 that the Lipschitz condition on $\nabla f$, needed for gradient descent to converge, can be ...
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2answers
30 views

Show that $f_n \rightarrow 0$ in $C([0, 1], \mathbb{R})$

I was given the following problem and was wondering if I was on the right track. Let $f_n(x) = \frac{1}{n} \frac{nx}{1 + nx}, \: 0 \le x \le 1$ Show that $f_n \rightarrow 0$ in $C([0, 1], \mathbb{R}...
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2answers
36 views

Alternating series example

I have a problem with rather simple example: $$\sum\limits_{n=0}^{\infty} (-1)^{n+1}\frac{10^n}{n!}$$ I have to tell if it's convergent. I know it is, but I don't know how to prove it. I was ...
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1answer
26 views

Does calculating the limit of a given real-valued (recursive) sequence already imply its convergence?

Assuming we are given a real-valued recursive sequence $(a_n)_{n \in \mathbb{N}}$ by its starting point $a_1$ and its recursive function $a_{n+1} = \varphi(a_n)$ EDIT: Since this question caused some ...
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4answers
74 views

Convergence of $\sum_{n=0}^{\infty}(-2)^{n^2}/n!$ [on hold]

click here to see my ratio solution$$\sum_{n=0}^\infty\frac{(-2)^{n^2}}{n!}$$ I am supposed to find out if this series is convergent, absolutely convergent or divergent. No test has given me ...
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1answer
34 views

difference between convergence along free ultrafilter and common convergence

Suppose $\mathcal{F}$ is any free ultrafilter on $\beta\mathbb{N}\setminus\mathbb{N}$,$(x_n)$ is a sequence of complex numbers. My question is:What is the deference between the limit along $\mathcal{...
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0answers
3 views

Discrete consensus power mismatch

I am finding it difficult to simulate 2 agents to reach consensus on matlab based on their power mismatch. can anyone help me solve it?enter image description here
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3answers
3k views

What is the formula for pi used in the Python decimal library?

(Don't be alarmed by the title; this is a question about mathematics, not programming.) In the documentation for the decimal module in the Python Standard Library, ...
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2answers
19 views

Proof for Convergence of Nested Sequences

With this being my first week learning about limits, their rules, and convergence/divergence, I received another question: $a_k$ is a convergent sequence with $\lim_{n\to \infty}(a_k) = a$ Let $p(x) ...
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1answer
21 views

Spectral radius equal to 1 and convergence

The following theorem is well-known: $$ \lim_k A^k = 0 \text{ if and only if } \rho(A)<1 $$ (see wiki for context and proofs). What if now $\rho(A)=1$ and $\lambda\neq -1$ for all $\lambda \in ...
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0answers
33 views

If $(x_{n})$ is sequence with $x_{n} \leq b$ for all $n$, then $\text{lim}_{n \to \infty} x_{n} \leq b$

If $(x_{n})$ as sequence with $x_{n} \leq b$ for all $n$, then $\text{lim}_{n \to \infty} x_{n} \leq b $ if limit exists. I see this fact being used very often and it seems fairly trivial but I am ...
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3answers
46 views

$f_n(x) = n^2x^n(1-x)^2 $ uniformly convergent? [on hold]

Let $$f_n(x) = n^2x^n(1-x)^2\quad \text{on $I = [0,1]$}$$ Find the pointwise limit and determine whether or not the convergence is uniform and provide reasoning. Hint: consider $f_n=(1-n^{-1})$. ...
0
votes
0answers
27 views

What is the radius of convergence of $\sum_{k\geq 0}3^{k^2}x^{k^2}$

What is the radius of convergence of $\sum_{k\geq 0}3^{k^2}x^{k^2}$. MY TRIAL Using Cauchy's root test $$\limsup\limits_{k\to\infty}\sqrt[k]{3^{k^2}}=\limsup\limits_{k\to\infty}3^k=\infty.$$ So, ...
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votes
2answers
56 views

Does $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}-\frac{2}3}$ converge or diverge?

Does this series converge or diverge? $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}-\frac{2}3}$$ I tried using the limit comparison test with $\frac{1}{\sqrt{n}}$, which diverges. $$\lim_{n\to\infty}{\...
4
votes
3answers
84 views

Convergence of Sum of Sequences

This week, I learned a bit more about limits, convergence and divergence. I was given a sum of two sequences and asked to tell whether or not it is convergent, and what its limit is: $a_n := (-1)^n +...
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2answers
69 views

How to show that if $\lim_{n\to\infty}(a_n) = a > 0$ then $\lim_{n\to\infty}(a_n)^{1/n} =1$ [on hold]

Let $(a_n)$ be a sequence such that $a_n > 0$ for all $n \in \mathbb{N}$ and $$\lim_{n\to\infty}(a_n) = a > 0$$ Show that: $$\lim_{n\to\infty} (a_n)^{1/n}=1$$ Any help or hints would be ...