Convergence of sequences and different modes of convergence.

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Convergence of a subsequence

I have read that : If $\sum_{n=1}^{\infty}a_n$ is a series whose terms form a positive monotone decreasing sequence $(a_n)$, then it converges and diverges with $$ \sum_{k=0}^{\infty}2^ka_{2^k} = ...
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1answer
81 views

Convergence of $\sum \frac{\lambda^y}{y!}$ to $e^\lambda$ [duplicate]

How can I show that $$ \sum_{y=0}^\infty \frac{λ^y}{y!} = e^λ $$? I only get so far as to show that the series will converge (by ratio test), but not further, I'm not even sure where to start.
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1answer
22 views

Convergence of products of sequences implies convergence of sequence itself

In my thesis I encountered the following problem for which I could not find a solution in literature: I have a sequence $(x_n)_{n\in\mathbb{N}}$ in a Banach space $F$ which has the following property: ...
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74 views

Prove $\{u_n\}$ converges to $\sqrt[3]{u_2^2 u_1}$ without using subsequences. [duplicate]

Prove that the sequence $\{u_n\}$ is defined by $0\lt u_1\lt u_2 \;\text {and} \;u_{n+2}=\sqrt {u_{n+1}u_n}\; \text{for}\; n\ge 1$, converges to $\sqrt[3]{u_1{u_2}^2}$. I did this using subsequences,...
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31 views

Close to a proof that every such equicontinuous sequence of functions converges uniformly.

Let $X$ be a compact metric space; let $Y$ be a normed space; let $\{ f_{n} \} \subset \mathscr{C}(X, Y)$ be equicontinuous; and let $f_{n} \to g$ pointwisely for some $g: X \to Y$. Claim: We have $f_{...
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1answer
32 views

Test for convergence - Log operation

∑ Log (n / n+ 1) I solved the above problem in two different ways. 1st Method = Log n - Log n+1 = Log n - Log n . Log 1 = Log n - Log n . 0 = Log n = Log ∞ = ∞ (Diverges) Method 2 = Log ( n / n + ...
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2answers
54 views

I know a function $f(x)$, how can I show it's equal to a known Fourier series, using geometric series

I have a function $$f(t) = \frac{1}{4}\left(\frac{3}{e^{i\,t}+3} -\frac{1}{3e^{i\,t}+2}\right)$$ I want to show that it's Fourier series is equal to $$f(t) = \frac{1}{4}\sum^\infty_{k=-\infty}\left(\...
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1answer
27 views

Is Taylor's series for cos uniformly convergent?

Is it true that Taylor's series for cos uniformly convergent (for all $\mathbb{R}$)?
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60 views

Monotonicity, boundaries and convergence of the sequence $ \left\{ \frac{a^n}{n!} \right\} $.

everyone. I have a doubt on the following question: Let $ \left\{ \frac{a^n}{n!} \right\}, n \in \mathbb{N} $ be a sequence of real numbers, where $ a $ is a positive real number. a) For what ...
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77 views

Proof convergency of series $a_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \ldots + \frac{1}{n!} $

I have used Cauchy and came to step where i have $\frac{1}{(n+1)!} + \frac{1}{(n+2)!} + \ldots + \frac{1}{(n+p)!} $ i cant find upper boundary $ \epsilon $ , hope you guys can help me. Thanks in ...
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3answers
44 views

convergence of sequence with factorials

i want to study wether the sequence $$u_n=\frac{1}{n!}\sum_{k=1}^nk!$$ converges or not. I didn't get the trick yet, do you have any clue?
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1answer
40 views

Proving a partial sum is convergent

Assume $a_n$>0 and lim($n^2a_n$) exists. I need to show that the sequence of partial sums of ($n^2a_n$) converges. I know that ($n^2a_n$) is bounded and increasing. I can also show that its sequence ...
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0answers
53 views

Showing partial sums converge based on sequence convergence

Assuming $a_n$$>$0 and that lim($n^2a_n$) exists, I need to show that $\sum$$a_n$ converges. My plan is to show that the sequence of partial sums, let's say $s_n$ of ($n^2a_n$) converges, and ...
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2answers
104 views

An example of a locally uniformly convergent sequence which does not converge uniformly

I'm looking for an example of a sequence of real-value functions $\left\{ f_{n}:\mathbb{R}\to\mathbb{R}\ |\ n\geq1\right\}$ which converges locally uniformly to some function $f$ on some interval $\...
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100 views

Find sum and absolute convergence of series

I am looking at: $$\sum_{n,m=1}^\infty \dfrac{1}{(n+m)!},$$ my task is to show that it is absolutely convergent and to find its sum. I have found the sum doing the following: $$\sum_{m,n=1}^\...
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1answer
95 views

Determine convergents for square root

All square roots can be represented as a continued fraction. The fraction can be calculated to $n$ terms (e.g. $\sqrt{2}$ is $[1; 2, 2, 2, 2...]$) So the continued fraction for $\sqrt{2}$ to $3$ ...
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1answer
41 views

Convergence problem in Gummel iterations

I am trying to solve two non-linear equations self-consistently in a Gummel loop. Sometimes (every once in a while), I get to a situation when the loop repeats itself with wrong solutions and a ...
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4answers
170 views

A question about sequences of integrable functions

Say $f_n$ is some sequence of Lebesgue integrable maps $[0,1] \to \mathbb R$ such that for all $x\in [0,1]$: $$ \lim_{n \to \infty} f_n (x) = 0$$ That is, the pointwise limit is the zero function. ...
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1answer
35 views

Can this series diverge?

Suppose we have a series $\sum x_n$ that is known to be convergent. Further, $(y_n)$ is a bounded sequence. Can $\sum x_ny_n$ diverge? If so, provide an example. If not, justify. I have been thinking ...
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1answer
75 views

Power series radius of convergence equal to r, nonzero for every number meeting conditions

I have been stuck on this question for a few hours now. It seems like it should be simple, but for some reason I can not figure out a solid proof. It seems like a simple series proof, but the $a$ is ...
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1answer
33 views

Sequence Convergence Proof

Show that if d is a number and x1, x2, x3, ... is a sequence which converges to the point c, then the sequence d · x1, d · x2, d · x3, ... converges to d · c. According to my text, the statement that ...
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1answer
106 views

If limit exists then its radius of convergence is this power series

I am confused as to how if $\lim_{n\to\infty} | c_{n} / c_{n+1}|$ exists, then it equals the radius of convergence of the power series $\sum_{n=0}^\infty c_{n}x^{n}$ How is this so?
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74 views

Series Divergence Proof

I'm asked to decide whether the following series converges or diverges: $$1-\frac{3}{4}+\frac{4}{6}-\frac{5}{8}+\frac{6}{10}-\frac{7}{12}+\cdots$$ So I first looked at $(a_n)=\frac{n+1}{2n}$. Then ...
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2answers
75 views

Prove or disprove - if $f(0)=f'(0)=0$ then $\sum_1^\infty f(1/n)$ converges

Prove or disprove - if $f(0)=f'(0)=0$ then $\sum_1^\infty f(1/n)$ is convergent. I've been trying to solve this problem for a long while. Any direction will be much appreciated.
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42 views

Euler-summability of a convergent sequence and its limit

In the book Finite Markov Chains by John G.Kemeny and J.Laurie Snell, the authors introduce the following concept of Euler-summability of a sequence: Having a sequence $(s_n)_{n \geq 0},$ we ...
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1answer
66 views

Is this subspace closed?(linear functions converges always to a linear function?)

We are working on functions that are in $L^2([-1,1])$, is the space of functions that are linear $f(x)=ax+b$, closed? I am not entirely sure of how to prove this. I mean, if I have a sequence of ...
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1answer
21 views

Showing convergence of this function defined by a series

Let $f: [0,\infty) \to \mathbb{R}$ be continuous and define $$ f_n(x) = \sum_{j=0}^\infty f(j\cdot 2^{-n}) \mathbb{I}_{[j\cdot 2^{-n}, (j+1)\cdot 2^{-n})}(x). $$ I'd like to show pointwise (or ...
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59 views

Understanding Error Convergence in Numerical Analysis

I'm taking an introductory Numerical Analysis course, and we recently talked about error convergence for iterative algorithms. As a precursor to the question, we denote the absolute error in the ...
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1answer
68 views

Does infinite repeated convolution with the same normal distribution converge?

According to Wikipedia, This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. ...
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43 views

Prove that the series $\displaystyle \sum_{n=1}^\infty a_n$ converges in $X$.

Let $\{a_n\}$ be a sequence in a Banach space $X$ such that $\displaystyle\sum_{n=1}^\infty \|a_n\|\lt \infty$. Prove that the series $\displaystyle \sum_{n=1}^\infty a_n$ converges in $X$. To show ...
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236 views

Convergence of $\sum a_{n}$,$\sum a_{n}^{2}$ and $\sum a_{n}^{4}$

A Convergent series of real numbers $\sum a_{n}$ is given , what can be said about the convergence of $\sum a_{n}^{2}$ and $\sum a_{n}^{4}$. Also , if only absolute convergence of $...
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0answers
25 views

Prove if $\sum_{n=0}^{\infty}a_n(x-x_o)^n$ is conditionally convergent in $x_1$, then $R=|x_1-x_0|$

Prove if $\displaystyle\sum_{n=0}^{\infty}a_n(x-x_o)^n$ is conditionally convergent at $x_1$, then $R=|x_1-x_0|$ ( $R$ is radius of convergence) First, According to Abel's Theorem. I know that: (1)...
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380 views

Infinite sum of reciprocal shifted Fibonacci numbers

I found on Wikipedia the following infinite sum : $$\sum_{k=0}^{\infty} \frac{1}{1+F_{2k+1}}=\frac{\sqrt{5}}{2}$$ There is no reference for this sum in the article and I couldn't find it ...
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38 views

Is $\sum_{n=1}^{\infty}n!(\frac{x}{n})^n$ convergent for $x\ge 0$?

Let $u_n=n!(\frac{x}{n})^n$ then $$\lim_{n\to \infty}\frac{u_{n+1}}{u_n}=\frac{x}{e}$$ So when $0\le x\lt e$, it's convergent. And I stuck in the case when $x=e$.
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60 views

How could I show that $\int_{0}^{\frac{\pi}{2}} \frac{\cos (nx)}{\cos x}dx$ converges to zero as $n\rightarrow \infty$?

Consider $$\int_{0}^{\frac{\pi}{2}} \frac{\cos (nx)}{\cos x}dx$$ From here, I can show that $$\int_{0}^{\frac{\pi}{2}} \frac{\cos (nx)}{\cos x}dx > \int_{0}^{\frac{\pi}{2}} \cos (nx)dx \...
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33 views

Sum of infinitely many power series and interchanging double summation

Fix $t\in\mathbb R$ and suppose that for each $v\in\mathbb N$, the power series $$\sum_{k=1}^{\infty} a_{v,k} t^k$$ is absolutely convergent. Suppose also that $$\sum_{v=1}^{\infty}|b_v|\left|\sum_{k=...
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1answer
50 views

Convergence of linear functional

I want to determine wether the functional $\varphi_n:\ell^2\to \mathbb{R}$ defined by $$\varphi_n(x)=\frac{1}{n}\sum_{k=1}^n\sqrt k x_k\quad x=(x_1,x_2,\dots)$$ converges in norm, or in weak sense. ...
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86 views

Is the space $\mathcal{H}=\{v\in H^1(\Omega):\Delta v\in H^1(\Omega)^*\}$ a Banach space?

Let $\Omega$ be a Lipschitz domain in $\Bbb R^n$, is the space $$\mathcal{H}=\{v\in H^1(\Omega):\Delta v\in H^1(\Omega)^*\}$$ a Hilbert space when endowed with the norm $\|\cdot\|_\mathcal{H} = \|u\|_{...
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135 views

If $(u_n)$ is bounded and $\lim u_n^2+u_n-u_{n+1}=0$ then $u_n \to 0$

Let $(u_n)$ be a real bounded sequence. Suppose that $\lim\limits_{n \to \infty} (u_n^2+u_n-u_{n+1})=0$. Prove that $u_n \to 0$. I was able to develop a prove, looking at the map $x \mapsto x^2+x$ ...
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1answer
22 views

Limit of sequence partially applied to a function

Let $(a_n)_{n\in\mathbb N}$, $(b_n)_{n\in\mathbb N}$, $(c_n)_{n\in\mathbb N}$ and $(d_n)_{n\in\mathbb N}$ be real-valued sequences and $f:\mathbb R\to \mathbb R$ monotone with $\lim_{x\to\infty}f(x)=\...
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1answer
24 views

Investigating uniform convergence of a sequence

I am trying to determine if the sequence $f_n$:= $\frac{x^{2n}}{1+x^{2n}}$ is uniformly convergent on $D_1:=[-q,q],0<q<1$, and $D_2:= (-\infty,r] \cup [r,\infty),r>1$. I have determined that ...
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38 views

Bounded derivative and Taylor's polynomial

Given function $f:\mathbb{R}\rightarrow \mathbb{R}$ of class $C^{\infty}$ such as $\forall_{n\ge2015}{\forall_{x\in\mathbb{R}}{|f^{(n)}(x)|\le7}}$ a)prove that sequence of functions $\{T_{n,f,0}\}$ ...
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27 views

Linear Operators and Convergence

I am struggling with this question from my Differential Equations course: If $T$ is a linear transformation on $\mathbb{R}^n$ with $||T-I||<1$, prove that $T$ is invertible and that the ...
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2answers
71 views

Proving that if $p_{n}$ converges then $|p_{n}| $ converges

EDIT1: Prove using the definition of a converging sequence in a metric space, that the convergence of the sequence $\left \{ p_{n} \right \}_{n=1}^{\infty}$ implies the convergence of the sequence $\...
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1answer
83 views

A sequence that converges weakly but not in the Cesàro sense

Let $H$ be a Hilbert space over $\mathbb{C}$ with inner product $\langle\cdot,\cdot\rangle$, and let $\{x_n\}_{n=1}^\infty\subseteq H$, $x\in H$. I'm using the following definitions: $\{x_n\}_{n=1}^\...
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0answers
38 views

Max function as bounded functions

I have an algebra of bounded functions $A$ that contains the constant functions and is closed under uniform convergence. We also have that if $f \in A$ then $|f| \in A$. I'm trying to show that if $f, ...
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1answer
44 views

When is it ok to use a sequential limit in place of a continuous limit?

I am working through some Lebesgue integral problems, and I've come across a few instances where I would like to use the dominated/monotone convergence theorems, but the limit is continuous, and I'm ...
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1answer
36 views

Divergent or convergent but how ??

I was to depict the convergence & divergence nature of the summation $\sum A_n$ where $A_n = (n^{1/n}-1)^k$ I was able to prove that when $k>1$ then $\sum A_n$ is converging and while $k<0$ ...
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4answers
93 views

Proving convergence of a series and then finding limit [duplicate]

I need to show that $\sqrt{2}$, $\sqrt{2+\sqrt{2}}$, $\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$ converges and find the limit. I started by defining the sequence by $x_1=\sqrt{2}$ and then $x_{n+1}=\sqrt{2+...
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1answer
253 views

Is the extended real line a metric space?

I've got a question reading the demonstration of the Theorem 3.2 in POMA of Rudin. Indeed, he says that every convergent sequence in a metric space is bounded. My question is: Is $\bar{\mathbb{R}}$...