Convergence of sequences and different modes of convergence.

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11
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1answer
164 views

Nested Radicals Involving Primes

How do you evaluate $\sqrt { 2+\sqrt { 3+\sqrt { 5+\sqrt { 7+\sqrt { 11+ \dots } } } } } $ ? This question appears to be rather difficult as there is no way to perfectly know what $p_{ n }$ is , ...
10
votes
8answers
617 views

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
votes
8answers
6k 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 ...
10
votes
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: $$\gamma = \lim_{n \rightarrow \infty }\left(\sum_{k=1}^n \frac{1}{k} - \ln(n)\right)$$ ...
10
votes
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
votes
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 ...
10
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2answers
747 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 ...
10
votes
1answer
185 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 ...
10
votes
4answers
230 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 ...
10
votes
1answer
4k views

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 ...
10
votes
3answers
376 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} ...
10
votes
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 ...
10
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3answers
1k views

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 ...
10
votes
2answers
514 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 ...
10
votes
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 ...
10
votes
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 ...
10
votes
3answers
269 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 ...
10
votes
2answers
4k views

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 ...
10
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1answer
440 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 ...
10
votes
3answers
3k views

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, ...
10
votes
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 ...
10
votes
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 ...
10
votes
1answer
609 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 = ...
10
votes
0answers
84 views

Are the fractional parts of powers of $\pi$ divergent?

Let us define $a_n$ as the fractional part of $\pi^n$. In other words, define $a_n=\pi^n-\lfloor \pi^n \rfloor$. Then, does the following limit exist? $$\lim_{n \to \infty}a_n$$Intuitively, it ...
10
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0answers
114 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 ...
9
votes
4answers
689 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 ...
9
votes
2answers
120 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 ...
9
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2answers
518 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 ...
9
votes
3answers
2k views

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.
9
votes
2answers
3k views

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 ...
9
votes
3answers
898 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 ...
9
votes
2answers
605 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 ...
9
votes
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 ...
9
votes
1answer
363 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 ...
9
votes
3answers
654 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 ...
9
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3answers
746 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 ...
9
votes
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 ...
9
votes
1answer
192 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 ...
9
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2answers
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: ...
9
votes
2answers
277 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 $\bf CM$. ...
9
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2answers
277 views

methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$

As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which ...
9
votes
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 ...
9
votes
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 ...
9
votes
1answer
341 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? $$\sum_{n=1}^\infty \cos{p_n}$$
9
votes
1answer
147 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 ...
9
votes
1answer
342 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 $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
9
votes
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 ...
9
votes
1answer
65 views

Is it possible to study the properties of sequences by studying the family of polynomials generated with the elements as coefficients?

Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows: $p_0 = a_0$ $p_1 = a_0x+a_1$ $p_2 = a_0x^2+a_1x+a_2$ $p_n = ...
9
votes
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 ...
9
votes
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 ...