Convergence of sequences and different modes of convergence.

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Convergence of the sequence $a_n=\int_0^1{nx^{n-1}\over 1+x}dx$ [duplicate]

How to prove the following sequence converges to $0.5$ ? $$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$ What I have tried: I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n ...
10
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4answers
448 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: $$\gamma = \lim_{n \rightarrow \infty }\left(\sum_{k=1}^n \frac{1}{k} - \ln(n)\right)$$ ...
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5answers
223 views

Prove $\lim\limits_{n\to\infty}\int_0^1(\cos \frac1x)^n\mathrm dx=0$

Prove $$\lim_{n\to\infty}\int_0^1 \left(\cos{\frac{1}{x}} \right)^n\mathrm dx=0$$ I tried, but failed. Any help will be appreciated. At most points $(\cos 1/x)^n\to 0$, but how can I prove that the ...
10
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3answers
4k views

$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$

Please help me check, if $f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n + g_n$ uniformly converge to $f+g$ $f_n$ uniformly converge to $f$ and $g_n$ uniformly ...
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2answers
734 views

Stone-Čech compactifications and limits of sequences

I've been working on some old prelims from my university when they used to just be on point-set topology. We don't cover a couple of the topics so I've been teaching myself some of the material, one ...
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1answer
182 views

$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.

Let $(X, M, m)$ be an arbitrary measure space. Let $f_n, f \in L^1_m(X)$. Assume that $$\int_X |f_n - f| \, dm \leq \frac{1}{n^2} \text{ for all }n \geq 1. $$ Then I want to show that $f_n ...
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4answers
229 views

Compute $\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$

Compute \begin{equation} \lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx \end{equation} According to Wolfram Alpha, the limit is zero. I tried to make substitution ...
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1answer
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Convergence of Ratio Test implies Convergence of the Root Test

In Elias Stein and Rami Shakarchi's Complex Analysis textbook, we have the following exercise: Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of complex numbers such that ...
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3answers
372 views

For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?

The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges. What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} ...
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5answers
188 views

Convergence of a sequence $c_n$

Suppose that $(a_n)$ and $(b_n)$ be sequences such that $\lim (a_n)=0$ and $\displaystyle \lim \left( \sum_{i=1}^n b_i \right)$ exists. Define $c_n = a_1 b_n + a_2 b_{n-1} + \dots + a_n b_1$. Prove ...
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3answers
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Does $f_{n}(x)=n\cos^n x \sin x$ uniformly converge for $x \in [0,\frac{\pi}{2}]$?

I want to check whether the following function is uniformly converges: $f_n(x)=n\cos^nx\sin x$ for $x \in \left[0,\frac{\pi}{2} \right]$. I proved that the $\lim \limits_{n \to \infty}f_{n}(x)=0$ for ...
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2answers
501 views

Convergence/Divergence of $\int_e^\infty \frac{\sin x}{x \ln x}\;dx$

I am currently doing some project and during the course of it I need to get an answer to the following: Does $\displaystyle \int_e^\infty \frac{\sin x}{x \ln x}\;dx$ converge/ absolutely ...
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1answer
463 views

A question from the dreams realm

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a function (not necessarily continuous). Let $\phi_0(x)=\phi(x)$ and $\forall k\in\mathbb{N},\phi_{k+1}(x)=\phi(x\cdot\phi_k(x))$. 1. Let ...
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2answers
136 views

How to prove that $\sum_{n=1}^{\infty}\frac{1!+2!+\cdots+n!}{(2n)!}$ converges?

Show that this series converges: $$\sum_{n=1}^{\infty}\dfrac{1!+2!+\cdots+n!}{(2n)!}$$ My solution: this series converges since $$1!+2!+\cdots+n!\le n!+n!+\cdots+n!=n\cdot n!$$ and since ...
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3answers
266 views

Is the sequence defined by the recurrence $a_{n+2}=\frac{1}{a_{n+1}}+\frac{1}{a_n}$ convergent?

Let $a_0=1$, $a_1=1$ and $a_{n+2}=\frac1{a_{n+1}}+\frac1{a_n}$ for every natural number $n$. How can I prove that this sequence is convergent? I know that if it's convergent, it converges to ...
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2answers
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Proof that a sequence converges to a finite limit iff lim inf equals lim sup

This problem is purely for my own benefit, so I'd appreciate it if you offer help but don't spoil the proof for me. I've worked out the following solution, but I want to make sure that my reasoning ...
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1answer
436 views

Does Newton's method for inverting a series work?

Suppose we have $z=f(x)$ with $f$ an infinite series. We want to find $f^{-1}(z)=x$. Newton proposed the following method (as described in Dunham): First, we say $x=z+r$. We find $z=f(z+r)$, drop all ...
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3answers
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In what spaces does the Bolzano-Weierstrass theorem hold?

The Bolzano-Weierstrass theorem says that every bounded sequence in $\Bbb R^n$ contains a convergent subsequence. The proof in Wikipedia evidently doesn't go through for an infinite-dimensional space, ...
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2answers
207 views

The set of all permutations of indices such that the new series converges to the same limit forms a group?

Let $\sum_{i = 1}^{\infty} a_i = s \in \mathbb{C}$ be a convergent series of complex numbers. Then the set of all permutations $\sigma \in\operatorname{Perm}(\mathbb{N})$ such that ...
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1answer
587 views

Abel limit theorem

I would like to know if the Abel limit theorem works if the limit is infinite. Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...
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0answers
113 views

Does $\sum_n |\sin n|^{cn^2}$ converge?

So I recently asked a question about convergence of $\sum_n |\sin n|^{cn}$ for arbitrary $c > 0$ and it turns out that the terms of the series don't even converge, for any $c > 0$, so the series ...
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4answers
687 views

Please explain how Conditionally Convergent can be valid?

I understand the basic idea of Conditionally Convergent (some infinitely long series can be made to converge to any value by reordering the series). I just do not understand how this could possibly be ...
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8answers
5k views

Series that converge to $\pi$ quickly

I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does ...
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2answers
517 views

convergence of $\sum \limits_{n=1}^{\infty }\bigl\{ \frac {1\cdot3 \cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \frac {4n+3} {2n+2}\bigr\} ^{2}$

I am investigating the convergence of $$\begin{split}\sum _{n=1}^{\infty }\left\{ \dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \dfrac {4n+3} {2n+2}\right\} ^{2} &= \sum _{n=1}^{\infty ...
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2answers
118 views

If $\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = 0$, can $\sum_0^\infty a_n$ be rational?

If a nonzero sequence of rationals $$a_0, a_1 \dots a_n$$ "decays fast" in the sense that $\lim_{n \rightarrow \infty} a_{n+1}/a_n = 0$, can the series converge to a rational number? That is, can ...
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3answers
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Check convergence of $\sum^{\infty}_{n=1} \frac{1}{(\ln\ln n)^{\ln n}}$

Check convergence of $$\sum^{\infty}_{n=1}\frac{1}{(\ln \ln n)^{\ln n}}.$$ Please verify my solution below.
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2answers
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Does $\sum{\frac{\sin{(nx)}}{n}}$ converge uniformly for all $x$ in $[0,2\pi]$

This question arises because of a problem I was doing (Bartle 3rd edition, section 9.4 problem 3). It was like this. Given $a_n$ a decreasing sequence of positive numbers and suppose that ...
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3answers
890 views

How do I manipulate the sum of all natural numbers to make it converge to an arbitrary number?

I just found out that the Riemann Series Theorem lets us do the following: $$\sum_{i=1}^\infty{i}=-\frac{1}{12}$$But it also says (at least according to the wikipedia page on the subject) that a ...
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2answers
597 views

Convergence of $\sum _{k=1}^\infty \sin \left(\sqrt{k}\right)/k$

This was asked at an oral examination. Does the series $\displaystyle \sum _{k\geq1}\frac{\sin\left(\sqrt{k}\right)}{k}$ converge ? After playing with Mathematica, it's very likely it ...
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1answer
281 views

Is there a $k$ such that $a_n=\frac{n^k!}{(n^k!!)^2}$ converges?

Lately I have been playing around with the sequence $$a_n(k) := \frac{n^k!}{(n^k!!)^2}.$$ For $k=1$, it does not look much like it converges. I don't know $k=2$ it converges, but it doesn't really ...
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1answer
350 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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3answers
623 views

Nets and compactness in topological spaces.

I am reading Kelley’s book on general topology. There are a few statements on nets there (chapter 2), but the characterization of compact sets in the language of nets is not given. How should we prove ...
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3answers
738 views

Kummer's test - Calculus, Apostol, 10.16 #15.

I want to prove the following: Let $\{ a_n \}$ and $\{ b_n \}$ be two sequences with $a_n>0$ and $b_n>0$ for all $n \geq N$, and let $$c_n = b_n - \frac{a_{n+1}b_{n+1}}{a_n}$$ Then If there ...
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1answer
3k views

L'Hopital's rule and series convergence

I teach freshman calculus, and have recently been discussing series. In one question on a recent test, I asked whether $\sum\frac{n^2}{n^3+1}$ converges or diverges. One student got the correct ...
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1answer
187 views

How to show that the series of $\frac{\sin(n)}{\log(n)}$ converges?

Edit: I am seeking a solution that uses only calculus and real analysis methods -- not complex analysis. This is an old advanced calculus exam question, and I think we are not allowed to use any ...
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454 views

How prove this integral $\int_{0}^{\infty}f^{\alpha}(x)dx,\alpha>1$ is convergent

Question: let the function $f(x)\ge 0$,and such $$f'(x)\le\dfrac{1}{2},\forall x\ge 0$$ and this integral $\displaystyle\int_{0}^{\infty}f(x)dx$ is convergent. show that: ...
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2answers
274 views

Compact maps problem in Lax

In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...
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1answer
232 views

Prove the divergence of a particular series, given that another series diverges

Suppose $\{a_i\}_{i\in\mathbb N}$ is an increasing sequence of positive real numbers such that $$\sum_{n=1}^\infty\frac{1}{a_n}=+\infty.\tag{1}$$ Then I have to show that also ...
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1answer
249 views

Matrix algorithm convergence

Suppose I start with a $n \times n$ matrix of zeros and ones: $$ \begin{bmatrix} 0 & 0 & 0 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 ...
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1answer
340 views

Sum of cosines of primes

Let $p_n$ be the nth prime number, $p_1=2,p_2=3,p_3=5,\ldots$ How to prove this series converges/diverges? $$\sum_{n=1}^\infty \cos{p_n}$$
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146 views

Characterization for the convergence of a series

Problem. Let $X$ be a topological spaces which is compact and Hausdorff, $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$, and suppose there exists a sequence $\{x_n\}_{n\in\mathbb N}\subset X$, such that ...
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1answer
337 views

Does a convergent power series on a closed disk always converge uniformly?

If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
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1answer
246 views

Convergence in topologies

Let $\tau_1\subseteq \tau_2$ be two Hausdorff regular topologies in an infinite set $X$ such that the convergences of sequences in $\tau_1$ and $\tau_2$ coincide (they have the same convergent ...
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1answer
72 views

Are there certain conditions that $a_n$ must meet in order for this series to converge?

Suppose we have an alternating series of the form $$\sum_{n=1}^\infty \frac{(-1)^{a_n}}{n}$$ We know this converges for some basic cases, like when $a_n=n$ (the sum evaluating to $-\ln2$) or for ...
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2answers
249 views

How will the limit $\lim \sin(A^{n})$ behave for $|A| > 1$?

While answering to the following question Problem. Find the range of $x$ for which $\displaystyle \sum_{n=0}^{\infty} (-1)^{n} \sin(x^{-n})$ converges. Unlike some careless answerers who merely ...
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1answer
393 views

Convergence in $L^1$ space

Suppose that $f_{n}$ is a sequence of measurable functions, in a finite measure space, $f_{n}\to f $ in $m$-measure and that there exists $g$ in $L^1$ such that $\vert f_n\vert \le g$. Prove that ...
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1answer
1k views

Proof of a theorem of Cauchy's on the convergence of an infinite product

Well it is relatively well known that the condition for absolute convergence is given by the following theorem: In order that the infinite product $\prod _{n=1}^{\infty }\left( 1+a_{n}\right) $ may be ...
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2answers
317 views

Paradox or error in design?

Currently I'm writing a homework for my school. I've made an experiment built this way: There is a laser pointed at a half reflecting mirror which reflects 50% at a wall. The other half cross the ...
8
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7answers
292 views

Convergence of the sequence $(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$

I have a sequence $(a_n)$ where for each natural number $n$, $$a_n = (1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$$ and I want to find its limit as $n\to\infty$. I obviously couldn't ...
8
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9answers
14k views

Fastest Square Root Algorithm

What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...