For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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0answers
32 views

Different representation of $f(n) = \sum_{d|n; \ \sqrt n\le d \le n}(-1)^d$

I am looking for a different way to calculate the following sum where $d,n\in \mathbb N$: $$f(n) = \sum_{d|n; \ \sqrt n\le d \le n}(-1)^d$$ Here are some example results for different values of n ...
0
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1answer
34 views

Example of a sequence with at least 3 limit points [closed]

What is an example of a sequence that has at least 3 limit points?
2
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3answers
60 views

How to evaluate these two integrals about hyperbolic functions?

While I was calculating the two integrals below \begin{align*} \mathcal{I}&=\int_{0}^{\infty }\frac{\cos x}{1+\cosh x-\sinh x}\mathrm{d}x\\ \mathcal{J}&=\int_{0}^{\infty }\frac{\sin x}{1+\cosh ...
0
votes
1answer
32 views

Expansion of $S_n$ is less than $T_n$ [duplicate]

Let $S_n=\{(1+\frac{1}{n})^n\}$ and $T_n=\{1+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}\}$ I am trying to prove that $\lim_{n\rightarrow \infty} S_n > \lim_{n\rightarrow \infty} T_n$ Now I ...
1
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1answer
55 views

Can you 'take $\limsup$' in both sides of an inequality?

I'm reading the proof for $(1)$ of this paper, and I can't get the hang of how the author concludes the "hence we have that $L-\epsilon<\limsup b_n<L+\epsilon$", could anybody explain this? I ...
3
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1answer
44 views

Could I do this to an infinite series?

If a had two series like so: $$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} i + \sum^{\infty}_{k=1} k $$ Is it logical for me to say: $$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} i = ...
0
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0answers
16 views

Dual space and norm to $X=c_0\times l^1$, solution check

$$X=c_0\times l^1,\ \|(x,y)\|=\|x^\|_{\infty}+\|y\|_1$$ It is clear, that the dual space is isomorphic to $l^1\times l^\infty$ and the functional $x^*(x)$ is defined as ...
0
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1answer
34 views

Find the limit of the sequence $y_n$ defined by $x_{n}$

The sequence is defined as : $y_0 = x_0$ and $y_n = x_n - \alpha x_{n-1}$ where $ -1 \lt \alpha \lt 1 $ Also $ \lim\limits_{n\to \infty}{y_n}=b $ Find $ \lim\limits_{n\to \infty} {x_n}$ In my book ...
2
votes
1answer
50 views

Proof for sequence Let $x_n=(-1)^n\frac{n^2}{3^n}$

Let $x_n=(-1)^n\frac{n^2}{3^n}$ be a real sequence. It is claimed that $x_n \rightarrow 0$ for $n\ge4$. New to real analysis, I am having problems with the correct form for the proof. Let me try ...
3
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1answer
64 views

Convergence of $\sum_{n=1}^\infty \sqrt[n]{2}-1$

I'm trying to determine whether $$\sum_{n=1}^\infty \left ( \sqrt[n]{2}-1\right )$$ converges or diverges. Ratio, root, nth term, etc tests are either inconclusive or too difficult to simplify. I ...
0
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0answers
41 views

Convergence for trigonometric series

Does the following series converge or diverge? $$ s_{n}=\sum_{k=1}^{n}\cos\left(\frac{\pi k}{2}\right)\frac{k}{k+1000}\frac{1}{\sqrt{k}}, \text{ for n=1,2,}\ldots$$ I tried root test and ratio test, ...
0
votes
1answer
17 views

Recurrence relations with multiplication

I have a recurrence relation and I am not quite sure if I am solving it correctly. The relation is this: $t_n = 2nt_{n-1}$ where $t_0 = 1$ Here is how I went about solving this: First step is to ...
0
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1answer
23 views

Proving the impossibility of a particular binary sequence

Let $\Omega = \{0,1\}^{\mathbb{N}}$. My question is as follows. Can there exist an $\omega \in \Omega$ such that $$\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \omega_{k+a} = \frac{1}{2} \qquad ...
1
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0answers
24 views

Show $\limsup S_k=y$ and $\liminf S_k=x$ from rearrangement of series

I'm in a process of proving the last part of Riemann's Theorem on conditionally convergent series. The theorem states: Let $\sum a_n$ be a conditionally convergent series with real-valued terms. ...
4
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3answers
71 views

A result of equation $y^2+1=x^p$ where $p$ is odd prime.

Example 2.4.4 page 23 of the book "Problems of algebraic number theory" by R. Murty is about solving equation $y^2+1=x^p$ where $p$ is odd prime and $x,y\in \mathbb{Z}$. Solving this example lead to ...
2
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1answer
47 views

Double sequence, if $(x_m)_m$ and $(y_n)_n$ converge, then they have the same limit?

Let $(a_{m, n})_{m, n}$ be a "double sequence" of real numbers; that is, for every pair $(m, n) \in \mathbb{N} \times \mathbb{N}$, $a_{m, n}$ is a real number. Suppose that, for every $m$, $(a_{m, ...
-1
votes
2answers
39 views

Sequence $sin(\alpha * n)$ limit problem

I need help about this problem. I tried to solve($\alpha$=0) but I have no idea how to check for every $\alpha$. Problem: Find for which $\alpha \in \mathbb{R}$ this sequence has a limit. ...
1
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2answers
56 views

Double sequence, two sequences converge, but to different limits?

Let $(a_{m, n})_{m, n}$ be a "double sequence" of real numbers; that is, for every pair $(m, n) \in \mathbb{N} \times \mathbb{N}$, $a_{m, n}$ is a real number. Suppose that, for every $m$, $(a_{m, ...
-1
votes
4answers
60 views

Example of two sequences $(a_n)$ and $(b_n)$ such that both of them are bounded, neither of them is convergent, but $(a_n + b_n)$ is convergent? [duplicate]

What is an example of two sequences $(a_n)$ and $(b_n)$ such that both of them are bounded, neither of them is convergent, but $(a_n + b_n)$ is convergent?
0
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1answer
27 views

Non-infinite geometric sum; does not start at 0 or 1

It's bee a long time since I've worked with sums and series, so even simple examples like this one are giving me trouble: $\sum_{i=4}^N \left(5\right)^i$ Can I get some guidance on series like this? ...
6
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1answer
54 views

What is $\bigcap_{n \in \mathbb{N}} \left(0, {1\over n}\right)$?

What is$$\bigcap_{n \in \mathbb{N}} \left(0, {1\over n}\right)?$$I suspect it is the empty set, and we would see this by using the Archimedean property of $\mathbb{R}$ or something like that, but I ...
8
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4answers
74 views

If $ A=\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+…+\frac{1}{100\sqrt{99}}\;,$ Then $\lfloor A \rfloor =$

If $\displaystyle A=\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+.........+\frac{1}{100\sqrt{99}}\;,$ Then $\lfloor A \rfloor =$ Where $\lfloor x \rfloor$ represent floor ...
-1
votes
0answers
23 views

Uniform Convergence for Digamma function sum representation

I am dealing with the following summation: $$ \frac{1}{\psi_{(1)}(1)} \sum_{n=0}^{\infty} \frac{1}{(1+n-x)(1+n-y)} = \frac{-1}{\psi_{(1)}(1)}\frac{1}{x-y} \left( \psi_{(0)}(1-x) - \psi_{(0)}(1-y) ...
0
votes
2answers
16 views

Given two real numbers a and b and a<b, can we find a sequence $x_n$ which converges to a and $a<x_n<b$ ? Why? [closed]

I think We can use the Archimedean property but this will only give me a natural number bigger than a...
1
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2answers
41 views

Bounded sequence $a_n=\sqrt{4+2 \sqrt{4+\cdots+2 \sqrt{4+2 \sqrt{4+2 \sqrt{4+4}}}}}$

Let $$a_n=\sqrt{4+2 \sqrt{4+\cdots+2 \sqrt{4+2 \sqrt{4+2 \sqrt{4+4}}}}}$$ the sign $\sqrt{}$ occurs $n$ times. a) Prove, that $a_n< \sqrt{5}+1$ for all $n$. b) Find $\lim_{n\rightarrow \infty } ...
0
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0answers
14 views

Given $f:[a,b] \rightarrow R$, if $x'$ is a local minimum of $f$ and $x'<b$ then there exists a sequence $x_n$ converging to $x'$ with $x'<x_n<b$

I'm trying to understand the demostration of the folowing lemma: Is a function $f:[a,b] \rightarrow R$ differentiable in a local extrema $x'$ then $f'(x')=0$ Demostration: $x'$ is a local ...
0
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0answers
20 views

Is it possible to replace condition (c) by another condition for the case of sequences with finite number of elements

Definition: $(u_{k})_{k≥1}$ and $(v_{k})_{k≥1}$ are two adjacent sequences with the common limit $a$, if: (a) $(v_{k})_{k≥1}$ is decreasing (b) $(u_{k})_{k≥1}$ is increasing (c) $\lim \limits_{k\to ...
3
votes
2answers
38 views

True or false.If the series converges for x=1.1, then it converges for x=7

I saw this question in a previous year test and it seemed pretty simple, and that can often mean that I am missing something. If the series $$\sum_{n=0}^{\infty}a_n(x-3)^n$$ converges for $x=-1.1$, ...
3
votes
3answers
48 views

Sublimit $\mathbb{N}$ of sequence.

Problem: Is there sequence that sublimit are $\mathbb{N}$? If it's eqsitist prove this. I try to solve this problem by guessing what type of sequence need to be. For example: $a_n=(-1)^n$ has ...
2
votes
1answer
30 views

Sequence $a_1=n, a_{k+1}=2a_k-[\sqrt{a_k} ]^2$

Find all positive integers $n$ for which the sequence $$a_1=n, a_{k+1}=2a_k-[\sqrt{a_k} ]^2 $$ for all $k \ge 1,$ is periodic? Author: V. Yasinski
2
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1answer
44 views

How to evaluate $\displaystyle \int_{0}^{\pi }\theta \ln\tan\frac{\theta }{2}\mathrm{d}\theta$

I have some trouble in how to evaluate this integral: $$\int_{0}^{\pi }\theta \ln\tan\frac{\theta }{2}\mathrm{d}\theta$$ I think it maybe has another form $$\int_{0}^{\pi }\theta \ln\tan\frac{\theta ...
0
votes
1answer
26 views

Is the sequence $\Big \{\cos\Big(\dfrac 12 \tan^{-1}\big(-\dfrac n2\big)^n\Big)\Big \}$ monotone ?

Is the sequence $\Big \{\cos\Big(\dfrac 12 \tan^{-1}\big(-\dfrac n2\big)^n\Big)\Big \}$ monotone ? I can show that the sequence is convergent without any clue if it's monotone or not . Please help . ...
11
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1answer
303 views

How to prove this series about Fibonacci number?

How to prove this series: I have no idea where to start. $$\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$$ where $F_{1}=1,~F_{2}=1,~F_n=F_{n-1}+F_{n-2},~~n\geq 3$.
1
vote
1answer
31 views

Series generated by food donation

Not counting the fact that students need to take food from the shelves: Assume that I donate $1 $ pound of food and I get $3 $ of my students to donate, but they only donate $35\% $ of what I gave. ...
2
votes
3answers
88 views

Prove that $\lim\limits_{n\to\infty}1 + \frac{1}{1!} + \frac {1}{2!} + \cdots + \frac{1}{n!}\ge\lim\limits_{n\to\infty}(1+\frac{1}{n})^n$

So My professor assigned this question and I am really stuck on part B of the question. For $n \in \mathbb{N}$ let {$T_n$} = {$1 + \frac{1}{1!} + \frac {1}{2!} + \cdots + \frac{1}{n!}$}. (a) Prove ...
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1answer
30 views

Show that $a = \limsup_{n\to\infty} a_n$. [duplicate]

Suppose $a \in \mathbb{R}$ is such that: given any $ε>0$ there exists $n_0 \in \mathbb{N}$ such that $a_n \le a+\varepsilon$ for all $n \ge n_0$ there is $k\ge n_0$ for which ...
1
vote
1answer
31 views

Convergence of a sequence of Functions .

Let the function sequence $\{f_n\}$ be defined by $f_n(x)= x - 2 \exp(-nx) $ for $x \in \mathbb{R}$ . Now let $f :\mathbb{R} \rightarrow \mathbb{R} $ be defined by $f(x)= x-2I\{0\}(x)$ for $x \in ...
5
votes
3answers
3k views

Formula for the simple sequence 1, 2, 2, 3, 3, 4, 4, 5, 5, …

Given $n\in\mathbb{N}$, I need to get just enough more than half of it. For example (you can think this is : number of games $\rightarrow$ minimum turns to win) $$ 1 \rightarrow 1 $$ $$ 2 \rightarrow ...
1
vote
2answers
44 views

Find the generalized sum of $1+2(2)+3(2)^2+4(2^3)+…+n(2^{n-1})$ [duplicate]

Find the generalized sum of $1+2(2)+3(2)^2+4(2^3)+...+n(2^{n-1})$ I rewrote the above sequence into: $\sum_{k=1}^{n} k(2^{k-1})$. The sequence looks like a hybrid of the summation $\sum_{k=1}^{n} ...
2
votes
2answers
97 views

Will every subset of $R$ that is not bounded above contains a sequence that diverges?

Question: (a) Prove that every subset of $R$ that is not bounded above contains a sequence that diverges to infinity. (b) Prove that every unbounded subset of $R^d$ contains a sequence ($x_n$) with ...
2
votes
1answer
142 views

A series related to $\pi\approx 2\sqrt{1+\sqrt{2}}$

This question follows a suggestion by Tito Piezas in Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$? Q: Is there a series by Ramanujan that justifies the approximation ...
1
vote
5answers
68 views

If $\{x_n\}$ satisfies that $x_{n+1} - x_n$ goes to $0$, is $\{x_n\}$ a Cauchy sequence?

Since the definition of Cauchy sequence is: Understanding the definition of Cauchy sequence, I noticed we need an absolute value for $a_m-a_n$ in the definition so the statement would be false. But I ...
0
votes
0answers
18 views

Are sequences themselves metric spaces with the inherited metric?

I have been asked to show whether a sequence $(p_n)$ in $\mathbb{R}$ is a metric space with the inherited metric $d(x,y) = |x-y|$ It seemed to me at first to be a slightly odd question because we ...
2
votes
2answers
37 views

If $(x_n)$ is a Cauchy sequence, then it has a subsequence such that $\|x_{n_k} - x_{n_{k-1}}\| < 1/2^k$ [duplicate]

If $X$ is a normed linear space and $(x_n)$ is a Cauchy sequence in $X$, then $x_n$ has a subsequence $x_{n_k}$ which satisfies $$\|x_{n_k} - x_{n_{k-1}}\| < \frac 1 {2^k}$$ for every $k > 0$. ...
1
vote
2answers
85 views

Calculating square roots of positive numbers in a simple way

I'm learning math (for its applications) so this may be obvious. Is there a simple way to calculate the square root of a real number by using a function repeatedly that uses only plus, minus, add or ...
1
vote
2answers
41 views

Limit of power series with L'Hospital

Calculate the given limit: $$\lim_{x\to 0} \frac{1}{1-\cos(x^2)}\sum_{n=4}^\infty\ n^5x^n$$ First, I used Taylor Expansion (near $x=0$): $$1-\cos(x^2)\approx 0.5x^4$$ I'm now quite stuck with the ...
0
votes
0answers
46 views

Completing the proof of Riemann's Rearrangement Theorem [closed]

I'm in a process of proving Riemann's Theorem on conditionally convergent series as shown in the image, but stuck on proving the last part left for the reader: partial sums of this rearrangement have ...
2
votes
0answers
34 views

Find $\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}\frac{2k-1}{2^k}$ [duplicate]

What is the method finding the closed form of $\displaystyle\sum_{k=1}^{n}\frac{2k-1}{2^k}$?
0
votes
4answers
85 views

Series $\sum_\limits{n=0}^\infty \frac{(n+1)}{(n^3-7)}$

I would like to prove the series $\sum_\limits{n=0}^\infty\frac{(n+1)}{(n^3-7)}$ is convergent. I have tried the ratio test but it is inconclusive, what is the way to go here ? Thanks
2
votes
1answer
24 views

Problem completing proof that the Weierstrass $\wp$ function is uniformly convergent

I'm having trouble following Ahlfors logic in his text concerning the proof that the $\wp$ function is convergent. The proof comes down to whether the series $$\sum_{\omega\neq ...