Tagged Questions
0
votes
0answers
42 views
Uniform convergence of $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$.
Given that the series $\sum_{n=2}^{\infty}\dfrac{\mathrm{ln}(n)^{p}}{n}$ is convergent iff $p<-1$.
Show that $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$ is uniform convergent as $x \in ...
1
vote
2answers
44 views
Prove the convergence of the sequence.
Prove the convergence of the following sequence:
$$x_1 = \sqrt{a}$$
$$x_{n+1} = \sqrt{a + x_n}$$
2
votes
2answers
49 views
alternating series test for $\sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}$
I must determine if this series converges (using specifically the alternating series test) $$\sum_{n=4}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}.$$
I know the necessary and sufficient conditions are:
The ...
5
votes
2answers
48 views
“Nearly” Harmonic Series
It's well known that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0.
$$
What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$?
...
0
votes
1answer
54 views
Sequence version of L'Hospital's Rule
Consider two sequences $A_n$ and $B_n$ such that $B_n$ is monotonically decreasing and both $A_n$ and $B_n$ tend to zero.
Now let us consider the limits ...
0
votes
2answers
55 views
Definition of metastability
I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
3
votes
1answer
53 views
Fourier series $\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}$
Does anyone know the sum of Fourier series $$\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}?$$
I tried WA; it does not return a function.
1
vote
2answers
57 views
Showing that $ \sum \limits_{m=1}^{n} b_m x_{m-n}~\to~ ab$ as $n~\to~\infty$
If $x_n ~\to ~a$ as $n~ \to~ \infty$
Does:
$ \sum \limits_{m=1}^{n} b_m x_{n-m}~\to~ ab$ as $n~\to~\infty$?
$b_m ~\geq~0$ and $ b~\equiv~ \sum \limits_{m=1}^{\infty} b_m < \infty$
My attempt:
...
1
vote
0answers
70 views
Is zero a cluster point of $n\sin n$?
I have seen somewhere that if $0\le \alpha<1$, then zero is a cluster point of the sequence $n^\alpha \sin n, n=1,2,\cdots$.
My question is what if $\alpha=1$? Or $\alpha>1$?
3
votes
1answer
39 views
Show that sequence approaches fixed point of a function
Problem
Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
6
votes
1answer
44 views
Exercise on convergent series
I am stumped by the following exercise (3.24 in Biler--Witkowski's book "Problems in mathematical analysis"):
Let $f$ be a continuous, increasing function from $[0,+\infty]$ to itself. Show that ...
0
votes
2answers
67 views
A few problems on sup and nested intervals
I've been doing these 3 problems for a `proof´ oriented class, one i have found a solution (in fact has been asked here before but the threads are all closed), and checked a correct solution in the ...
5
votes
2answers
77 views
Determine the character of $\sum_{n=1}^{+\infty}{\frac{e^{i\theta n}}{n}}$
Determine the character of the following series:
$$\sum_{n=1}^{+\infty}{\frac{e^{i\theta n}}{n}}$$
where $\theta$ is a real parameter.
I try to divide the series with De Moivre' s formula:
...
1
vote
1answer
40 views
Show that the sum can be written as:
How can the left side be expressed as the right one?
$$\sum_{n\in \mathbb{N}} \frac{1}{n^2}-\sum_{n\in \mathbb{N}} \frac{1}{(2n)^2}=\sum_{n\in \mathbb{N\cup \{0\}}} \frac{1}{(2n+1)^2}$$
Thanks in ...
1
vote
0answers
43 views
Series expansion of $\sin(n\arccos(x))$
Let $n=2m+1$ be an odd positive integer. Is there a clever way to prove that the Maclaurin series of $\sin(n\arccos(x))$ is equal to $$(-1)^m\left(1+\sum_{k=1}^\infty ...
1
vote
1answer
26 views
On the limit of a Minkowski sum
Consider an open set $\mathcal{O} \subseteq \mathbb{R}^n$.
I am wondering if the set $$ \mathcal{S} := \lim_{k \rightarrow \infty} \ \mathcal{O} + \frac{1}{k} \mathbb{B} $$
is open or closed.
With ...
2
votes
1answer
130 views
Infinite series involving $\sqrt{n}$
I am looking for examples of infinite series, whose sum is expressed as distributions or known functions, with a $\sqrt{n}$ in each term, such as:
$$ \sum_{n=0}^{\infty} \sqrt{n} z^n, \quad ...
1
vote
1answer
24 views
Prove that for n~=n' sum is much smaller than the case with n=n'
Hi I want to prove that this summation is much smaller for $n\neq n'$ than for the case where $n=n'$. I have seen this fact with simulation results. But I don't know how to prove it in mathematics.
...
2
votes
1answer
48 views
show the result is right when $f\in C(\Bbb{R})$, but when $f$ is only Riemann integrable. Is it right?
Assume $f(x) \in C(\Bbb{R})$, and $$S_n(x)=\sum_{k=1}^{n}\frac{1}{n}f\left(x+\frac{k}{n}\right),n=1,2,\cdots,$$
show that: $\forall [a,b] \subset \Bbb{R}$ , $S_n$ converges uniformly
and if $f(x)$ ...
1
vote
0answers
31 views
Is there an algebra for divergent series summation operators?
Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : \{ Series \} \to \{ numbers \} $ be a regular, linear divergent series operator, which is either one of ...
-2
votes
2answers
47 views
Calculation Limit Of Some Sequence
Natural number $n$ and $\begin{cases} a_{1}=\cos 1 \\ a_{n}=\max(a_{n-1}, \cos n)\end{cases}$
Find the value of $\lim_{n\to\infty}a_{n}$.
2
votes
0answers
44 views
Discontinuous for rationals
Show that $f\left(x\right):=\sum_{n=1}^{\infty}\frac{\left\{nx\right\}}{n^2}$, where $\left\{nx\right\}$ is the fractional part of $nx$, is discontinuous for all rationals.
I guess it would be nice ...
0
votes
2answers
38 views
Closed and open sets
By regarding the real numbers with their natural topology, my textbook says, that:
$$ \left\{2 \pi n+\frac{1}{n}\;\bigg|\;n \in \mathbb{N} \right\}$$
is closed, which i understand, as every sequence ...
3
votes
1answer
45 views
An identity related to Legendre polynomials
Let $m$ be a positive integer. I believe the the following identity
$$1+\sum_{k=1}^m (-1)^k\frac{P(k,m)}{(2k)!}=(-1)^m\frac{2^{2m}(m!)^2}{(2m)!}$$
where $P(k,m)=\prod_{i=0}^{k-1} (2m-2i)(2m+2i+1)$, ...
2
votes
2answers
64 views
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.
0
votes
0answers
20 views
Sup and lim sup of a function defined by double series
It is unlikely that the following function has a closed form expression:
$$f(t)=\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)$$
...
2
votes
0answers
74 views
The Cantor Space and open, but not closed sets.
consider the space $\{0,1\}^{\mathbb{N}}$ of all infinite binary sequences, called the Cantor-Space. This space is metrizable with metric
$$
d(u,v) = 2^{-(r-1)} \qquad \textrm{ where } r = ...
2
votes
0answers
39 views
Closed form formula for a double series related to wave equation
Does anyone have a closed form formula for the double series
$$\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)?$$
This is related ...
0
votes
1answer
27 views
Proof of a special case of Banach's fixed point theorem
I have to prove the following special case of the theorem:
Let $f : I \to I$ be Lipschitz continuous on the closed (not bounded) interval $I=[0,\infty)$ with Lipschitz constant $L \lt 1$. Then $f$ ...
0
votes
1answer
61 views
Whether convergence in L2 norm implies convergence a.e.? [duplicate]
How to prove or disprove$$\lim_{n\to\infty}\|f_n-f\|=0\;\Rightarrow \;\lim_{n\to\infty}f_n(x)=f(x)\; a.e.?$$ Any hint is appreciated.
1
vote
1answer
47 views
Uniform convergence for $x\arctan(nx)$
I am to check the uniform convergence of this sequence of functions : $f_{n}(x) = x\arctan(nx)$ where $x \in \mathbb{R} $. I came to a conclusion that $f_{n}(x) \rightarrow \frac{\left|x\right|\pi}{2} ...
1
vote
1answer
24 views
Series formed by reciprocal of fixed points of a function
Let $S(f)=\{x:x>0,f(x)=x \}$, the series $\sum_{x \in S(f)}\frac{1}{x}$ converges for which function in the following?
(i) $\tan x$
(ii) $\tan x^2$
(iii) $\tan2x$
(iv) $\tan \sqrt x$
(v) ...
1
vote
0answers
157 views
Double Fourier Series $\cos(nx)\cos(my)$
Let $f(x,y) = xy$ on the square $[0, \pi]^2$. Find the Fourier cosine-cosine series of $f$.
I am working on this question with a group and one of us gets all the coefficients as zero. Is this correct ...
8
votes
1answer
145 views
The series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)$
Does anyone have a proof that
$$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)=\frac{\pi}{8}$$
0
votes
2answers
47 views
Find the Fourier series of $g (x) = f (x-a)$, where $f$ is $2\pi$-periodic and $a$ is a real number.
Find the Fourier series of $g (x) = f (x-a)$, where $f$ is $2\pi$-periodic and $a$ is a real number.
This is for real analysis so I cannot use Euler's formula to compute the Fourier coefficients.
1
vote
2answers
42 views
Sequence of functions - showing the series converges uniformly
Consider the sequence of functions
$$f_n:[0,1]\longrightarrow \mathbb{R},\qquad f_n:=(-1)^n(1-x)x^n,\qquad n\geq 0.$$
Show that the series $f(x):=\sum_{n=0}^{\infty}f_n(x)$ converges to $f$ ...
4
votes
1answer
50 views
Is there a real power series with radius of convergence 1 that converges uniformly on (−1,1)?
Is there a real power series with radius of convergence $1$ that converges uniformly on $(−1,1)$?
I am guessing the answer is yes, if we can construct a function with power series such that it ...
1
vote
1answer
63 views
Find the supremum of $\left ( n+1 \right )^{\frac{2}{n^2}}$
As in the topic, my task is to find supremum and infimum of a given set $$f(n):=\left ( n+1 \right )^{\frac{2}{n^2}}, n\in \mathbb{N}$$What is funny, I managed to do this task few weeks ago and I ...
0
votes
0answers
47 views
Simple proof of convergence of the series $ \sum_{n=1}^\infty \frac{1}{n^2}$ [duplicate]
How can I simply prove that the series
$$ \sum_{n=1}^\infty \frac{1}{n^2}$$
converges to $ \frac{\pi^2}{6}$ ?
1
vote
5answers
160 views
Use Cauchy product to find a power series represenitation of $1 \over {(1-x)^3}$
Use Cauchy product to find a power series represenitation of
$$1 \over {(1-x)^3}$$ which is valid in the interval $(-1,1)$.
Is it right to use the product of $1 \over {1-x}$ and $1 \over ...
4
votes
3answers
105 views
Proving Newton's Binomial Theorem
So, I've done most of the problem to this point, but just cannot figure out the last piece. I may just be missing the math skills needed to complete the proof (differential equations).
Problem (from ...
8
votes
2answers
155 views
Series $\sum\limits_{n=1}^\infty \frac{1}{\cosh(\pi n)}= \frac{1}{2} \left(\frac{\sqrt{\pi}}{\Gamma^2 \left( \frac{3}{4}\right)}-1\right)$
I was playing around with Mathematica and found that
$$\sum_{n=1}^\infty\frac1{\cosh(\pi n)} = \frac12\left(\frac{\sqrt{\pi}}{\Gamma \left(\tfrac34\right)^2}-1\right)$$
Does anybody know how to ...
1
vote
4answers
121 views
Convergence of the infinite series $ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$
How can I prove that for every $ x \notin \mathbb Z$ the series
$$ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$$
converges uniformly in a neighborhood of $ x $?
2
votes
3answers
65 views
Existence and value of $\lim_{n\to\infty} (\ln\frac{x}{n}+\sum_{k=1}^n \frac{1}{k+x})$ for $x>0$
Does the limit
$$W(x)=\lim_{n\to\infty} \left(\ln\frac{x}{n}+\sum_{k=1}^n \frac{1}{k+x} \right)$$
exist for all $x>0$? If so, what is the limit
$$\lim_{x\to\infty}W(x)?$$
1
vote
3answers
134 views
Definition of “not converging” and proving $(-1)^n$ does not converge to $1$.
Remember that a sequence $x_n, n = 1,2,3\cdots$ is said to converge to $x$ as $n → ∞$ if for all $ε > 0$ there exists an $N ∈ \mathbb{N}$ such that $|x_n − x| < ε$ for all $n ≥ N$.
(a) Complete ...
2
votes
1answer
46 views
Isn't $f_n(x)=\frac{{n^2}\ln x}{x^n}$ with $x\geq1$ uniformly convergent by $T$-test?
One Dr. showed to me that the function $f_n(x)=\dfrac{{n^2}\ln x}{x^n}$, $x\geq1$ is not uniformly convergent by $T$-test, but I showed it to converge to $0$ anyway.
$$\lim_{x\to \infty}T_n=\lim_{x\to ...
1
vote
2answers
77 views
Convergence of $\sum\limits^\infty _{k=0} a_k \sin(kx)+b_k \cos(kx)$
Ok, for the infinite series:
$$\sum^\infty _{k=0} a_k \sin(kx)+b_k \cos(kx)$$
How do I show that this converges on any finite interval if $\sum^\infty _{k=0} k(|a_k|+|b_k|)<\infty$?
Also, do the ...
4
votes
4answers
172 views
Does the series $\sum_{n=1}^\infty$ ${\sqrt{n+1}-\sqrt{n}}\over n$ converge or diverge?
Does this series converge or diverge?
$$\sum_{n=1}^\infty\frac{\sqrt{n+1}-\sqrt{n}}{n}$$
I tried the comparsion test with $\sum_{n=1}^\infty$ $1 \over n$ but this does not help.
I also tried to ...
0
votes
2answers
42 views
Find $\alpha$ such that series converge
Find all positive $\alpha$ 's such that the series
$\\
\sum_{n=1}^{\infty} ((n+1)^\alpha-n^\alpha)^2 $
is convergent.
Thanks beforehand.
7
votes
3answers
129 views
Series $\sum_{k=1}^\infty \left(\frac{1}{k}-\frac{1}{k+z} \right)$
If $z$ is an integer, the sum of the series
$$\sum_{k=1}^\infty \left(\frac{1}{k}-\frac{1}{k+z}\right)$$ is easy since it is a telescoping series. But if $z$ is a fraction, say $z=3/2$, I don't see ...
