Convergence of sequences and different modes of convergence.
30
votes
3answers
810 views
Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?
Does the following series converge or diverge? I would like to see a demonstration.
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}.
$$
I can see that:
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
21
votes
1answer
631 views
How to prove convergence of polynomials in $e$ (Euler's number)
These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$
This function goes to 2. I've calculated this with sage math ...
19
votes
2answers
545 views
How to prove that $\sum\limits_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum\limits_{k=1}^\infty \frac{1}{(a+k)^2}$ for $a>-1$?
A problem on my (last week's) real analysis homework boiled down to proving that, for $a>-1$,
$$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{1}{(a+k)^2}.$$ ...
18
votes
1answer
213 views
Are there always singularities at the edge of a disk of convergence?
Take a function that is analytic at 0 and consider its Maclaurin Series. Here are some examples I'll refer to:
$$\frac{1}{1-x} =\sum_{n=0}^\infty x^n$$
$$\frac{1}{1+x^2} ...
17
votes
7answers
962 views
“Why do I always get 1 when I keep hitting the square root button on my calculator?”
I asked myself this question when I was a young boy playing around with the calculator.
Today, I think I know the answer, but I'm not sure whether I'd be able to explain it to a child or layman ...
17
votes
6answers
289 views
Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-…$
Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-...$
The number of signs increases by one in each "block".
I have an idea. Group the series like ...
17
votes
3answers
431 views
Why can't $\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}>2$?
So we have$$\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}=x\\\sqrt{2}^x=x$$where $x=2$ heuristically seems like a good solution. However, $x=4$ seems like an equally good solution. I was told in passing ...
16
votes
4answers
367 views
A double sum $\sum \limits_{n=1}^{n=\infty}\left(\sum \limits_{k=n}^{k=n^2}\frac{1}{k^2}\right)$
How to evaluate $\displaystyle\sum_{n=1}^{n=\infty}\left(\sum_{k=n}^{k=n^2}\frac{1}{k^2}\right)$?
16
votes
0answers
650 views
Prove that sum is finite
Let $j \in \mathbb{N}$. Set
$$
a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!}
$$
and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.
Please help me to prove that the following sum is ...
15
votes
1answer
893 views
What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
I tried and got this
$$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$
$$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$
where $m$ is an ...
15
votes
3answers
395 views
A question on convergence of series
Suppose $(z_i)$ is a sequence of complex numbers such that $|z_i|\to 0$ strictly decreasing. If $(a_i)$ is a sequence of complex numbers that has the property that for any $n\in\mathbb{N}$
$$
...
14
votes
4answers
665 views
Why ${ \sum\limits_{n=1}^{\infty} \frac{1}{n} }$ is divergent , but ${ \sum\limits_{n=1}^{\infty} \frac{1}{n^2} }$ is convergent?
I don't understand why ${ \displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n} }$ is divergent, but ${ \displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n^2} }$ is convergent and its limit is equal to ...
14
votes
3answers
280 views
Does every sequence of rationals, whose sum is irrational, have a subsequence whose sum is rational
Assume we have a sequence of rational numbers $a=(a_n)$. Assume we have a summation function $S: \mathscr {L}^1 \mapsto \mathbb R, \ \ S(a)=\sum a_n$ ($\mathscr {L}^1$ is the sequence space whose sums ...
14
votes
2answers
3k views
Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges?
We know that $\displaystyle\zeta(2)=\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ and it converges.
Does there exists a bijective map $f:\mathbb{N} \to \mathbb{N}$ such that the ...
14
votes
5answers
280 views
Convergent or Divergent? $\sum_{n=1}^\infty\left(2^{\frac1{n}}-1\right)$
Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent?
$$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$
I can't think of anything to compare it against. The integral looks too hard:
...
14
votes
1answer
427 views
Examples of Taylor series with interesting convergence along the boundary of convergence?
In most standard examples of power series, the question of convergence along the boundary of convergence has one of several "simple" answers. (I am considering power series of a complex variable.)
...
14
votes
2answers
476 views
If $\lim_n f_n(x_n)=f(x)$ for every $x_n \to x$ then $f_n \to f$ uniformly on $[0,1]$?
This is a self-posed question, so I do not know the answer and I would like to know what do you think about.
Let $f,f_n:[0,1]\to \mathbb R$ be continuous functions. Assume that for every sequence ...
14
votes
1answer
135 views
Does $\sum_{n=1}^\infty n^{-1-|\sin n|^a}$ converge for some $a\in(0,1)$?
The divergence of the series $\sum_{n=1}^\infty n^{-1-|\sin n|}$ is proved here. An inmediate consequence is that if $a\ge1$ then $\sum_{n=1}^\infty n^{-1-|\sin n|^a}$ also diverges. My question is: ...
13
votes
3answers
515 views
Convergence\Divergence of $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$
Prove convergence\divergence of the series:
$$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$$
Here is what I have at the moment:
Method I
My first way uses a result that ...
13
votes
2answers
625 views
“Pseudo-Cauchy” sequences: are they also Cauchy?
I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample.
Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k ...
13
votes
5answers
479 views
Slowing down divergence 2
Let $f(x)$ and $g(x)$ be positive nondecreasing functions such that
$
\sum_{n>1} \frac1{f(n)} \text{ and } \sum_{n>1} \frac1{g(n)}
$
diverges.
(Why) must the series $$\sum_{n>1} ...
13
votes
1answer
538 views
Infinite tetration, convergence radius
I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework.
The problem is stated as follows. Assume we have an infinite ...
12
votes
1answer
325 views
A few counterexamples in the convergence of functions
I'm studying the various types of convergence for sequences of real valued functions defined on measure spaces: pointwise convergence a.e. , convergence in $L^p$ norm, weak convergence in $L^p$, and ...
12
votes
1answer
197 views
Showing a series is convergent. [duplicate]
Possible Duplicate:
Contest problem about convergent series
Let ${p}_{n}\in \mathbb{R} $ be positive for every $n$ and $\sum_{n=1}^{∞}\cfrac{1}{{p}_{n}}$ converges,
How do I show that ...
11
votes
2answers
245 views
Generalized Fibonacci Sequence Question
The Fibonacci Sequence is defined as the recurrence
$a(n)=a_{n-1}+a_{n-2}$ where $a(0)=0$ and $a(1)=1$. Today, I was bored so I considered the sequence $a(n)=\sqrt{a_{n-1}}+\sqrt{a_{n-2}}$. Ten ...
11
votes
1answer
118 views
Does $f_n \to 0$ in $L^1(\mathbb R^2)$ imply that $f_{n_k}(x,\cdot)\to 0$ in $L^1(\mathbb R)$ for almost every $x \in \mathbb R$?
I would like to know what you think about this question. It is a "self-posed" question: I formulated it while I was doing an exercise.
Suppose you have $(f_n)_{n\ \in \mathbb N}\subset ...
11
votes
1answer
636 views
Strong and weak convergence in $\ell^1$
Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
11
votes
2answers
86 views
Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
11
votes
2answers
137 views
About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$
On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
11
votes
1answer
82 views
Existence of a specific reordering bijection
Please consider a bijection $g:\mathbb{N}\rightarrow\mathbb{N}$ with following properties:
For all real series $(a_n)_{n\geq1}$, convergence of $\sum_{n=1}^{\infty}a_n$ implies convergence of ...
11
votes
1answer
206 views
Convergence of the series $\sum_{n=1}^\infty \frac{(2n)!!}{(2n+1)!!} $
Study the convergence of the next series: $$\sum_{n=1}^\infty \frac{(2n)!!}{(2n+1)!!} $$
My solution: since $$\frac{(2n)!!}{(2n+2)!!} \leq \frac{(2n)!!}{(2n+1)!!}$$
forall $n \in \mathbb{N}$ and ...
10
votes
5answers
439 views
Why does this process, when iterated, tend towards a certain number? (the golden ratio?)
Take any number $x$ (edit: x should be positive, heh)
Add 1 to it $x+1$
Find its reciprocal $1/(x+1)$
Repeat from 2
So, taking $x = 1$ to start:
1
2 (the + 1)
0.5 (the reciprocal)
1.5 (the + 1)
...
10
votes
2answers
265 views
Is $\frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ entire?
Let $z$ be a complex number. Is the alternating infinite series $ f(z) = \frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ an entire function ? Does it even converge ...
10
votes
3answers
329 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
313 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
3answers
601 views
Product of two power series
Say if I define a power series over some arbitrary field $F$ as
$$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$
Then can I say:
$$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i ...
10
votes
1answer
907 views
Uniform convergence of series $\sum\limits_{n=2}^\infty\frac{\sin n x}{n\log n}$
Using Dirichlet series test I've proves that series $\displaystyle\sum\limits_{n=2}^\infty\frac{\sin n x}{n\log n}$ converges for all $x\in\mathbb{R}$.
How to determine whether the series ...
10
votes
1answer
146 views
Does Newton's method for inverting a series work?
Suppose we have $z=f(x)$ with $f$ an infinite series. We want to find $f^{-1}(z)=x$. Newton proposed the following method (as described in Dunham):
First, we say $x=z+r$. We find $z=f(z+r)$, drop all ...
10
votes
1answer
162 views
Abel limit theorem
I would like to know if the Abel limit theorem works if the limit is infinite.
Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...
9
votes
7answers
216 views
Convergence of the sequence $(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$
I have a sequence $(a_n)$ where for each natural number $n$,
$$a_n = (1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$$ and I want to find its limit as $n\to\infty$.
I obviously couldn't ...
9
votes
3answers
570 views
Check convergence of $\sum^{\infty}_{n=1} \frac{1}{(\ln\ln n)^{\ln n}}$
Check convergence of
$$\sum^{\infty}_{n=1}\frac{1}{(\ln \ln n)^{\ln n}}.$$
Please verify my solution below.
9
votes
3answers
125 views
Evaluating $\lim \limits_{n\to \infty} \left( n \int_{0}^{\frac \pi 2} 1-\sqrt [n]{\sin x} \,\mathrm dx \right)$
Evaluate the following limit:
$$\lim \limits_{n\to \infty} \;\; n \int_{0}^{\frac \pi 2} \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx $$
I have done the problem .
How I solved is
First I ...
9
votes
5answers
135 views
Convergence of a sequence $c_n$
Suppose that $(a_n)$ and $(b_n)$ be sequences such that $\lim (a_n)=0$ and $\displaystyle \lim \left( \sum_{i=1}^n b_i \right)$ exists. Define $c_n = a_1 b_n + a_2 b_{n-1} + \dots + a_n b_1$. Prove ...
9
votes
1answer
176 views
Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$
Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$
converges?
I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
9
votes
1answer
206 views
Prove the divergence of a particular series, given that another series diverges
Suppose $\{a_i\}_{i\in\mathbb N}$ is an increasing sequence of positive real numbers such that $$\sum_{n=1}^\infty\frac{1}{a_n}=+\infty.\tag{1}$$
Then I have to show that also ...
9
votes
1answer
141 views
Matrix algorithm convergence
Suppose I start with a $n \times n$ matrix of zeros and ones:
$$
\begin{bmatrix}
0 & 0 & 0 & 1 & 1\\
1 & 1 & 1 & 1 & 1\\
1 & 1 & 1 & 1 & 1\\
1 ...
9
votes
1answer
189 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
4answers
81 views
$a_{n+f(n)}-a_{n}\rightarrow0$ implies convergence?
$(a_{n})$ is a sequence of reals. Say $a_{n+f(n)}-a_{n}$ tends to 0 as n tends to infinity for every function f from the positive integers to the positive integers. Does this imply that $a_{n}$ is ...
9
votes
3answers
435 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, ...
8
votes
4answers
465 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 ...
