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

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0
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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
vote
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
votes
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
votes
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
votes
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
votes
1answer
304 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
89 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 ...
-1
votes
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
45 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
150 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
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5answers
69 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
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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
86 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
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0answers
48 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
86 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
25 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 ...
2
votes
1answer
53 views

Expected number of dice rolls of an unfair dice to roll every side equally many sides

I am having trouble with solving the following problem: The probability that a $d$-sided dice lands on its $k$th side is equal to $p_k$ for $k\in \{k\in\mathbb{N},k≤d\}$ and $p_1+p_2+p_3+...+p_d=1$. ...
3
votes
1answer
38 views

Is there a sequence of with every other sequence as a subsequence?

Let's say we have a set $S$. Is there a sequence $u:\mathbb N \to S$ such that every other sequence $v$ is a subsequence of $u$? Here is what I have so far: If $S=\emptyset$, then no (there are no ...
1
vote
5answers
167 views

Is the sequence $0,1,0,0,1,0,0,0,1,0,0,0,0,1,\ldots$ convergent? [on hold]

The sequence $0,1,0,0,1,0,0,0,1,0,0,0,0,1,\ldots$ is divergent according to a video I watched on WebAssign but the comment was very informal. Could someone provide a hint about the formal proof that ...
1
vote
0answers
52 views

Why the differential equations have a wave behavior?

The differential equation for string: $$\frac{1}{c^2} \frac{\partial^2 f}{\partial t^2}=\frac{\partial^2 f}{\partial x^2} \tag{1}$$ I have inital condition: $$f(x)=\begin{cases}20x, & 0\le ...
-2
votes
1answer
36 views

Is the set of $x$ such that the series $\sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}$ converges, bounded?

Is the following set bounded : $\{x\in \Bbb R:\sum _{n=0}^\infty \dfrac{(-1)^n x^{2n+1}}{(2n+1)!}$ is convergent $ \}$. I found the expansion of $\sum _{n=0}^\infty \dfrac{(-1)^n x^{2n+1}}{(2n+1)!}$ ...
3
votes
3answers
70 views

limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$

Problem: Find with proof limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ with $a_1=1$ or show that the limit does not exist. My attempt: I have failed to determine the existence. However if the ...
2
votes
0answers
35 views

Find the sequences $a_n$, such that $\sum_{n=1}^\infty\frac{1}{n^{2/(2-a_n)}}<\infty$.

Question: How much slow a positive sequence $a_n$ must converge to zero in order that $$\tag{1}\displaystyle\sum_{n=1}^\infty\frac{1}{n^{2/(2-a_n)}}<\infty.$$ I alread found that ...
1
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3answers
52 views

Find $\lim\limits_{n\to\infty}\left(\frac{1}{\sqrt{n}}\sum\limits_{k=1}^{n}\frac{1}{\sqrt{2k}+\sqrt{2k+2}}\right)$

I don't know how to find the sum of $\sum\limits_{k=1}^{n}\frac{1}{\sqrt{2k}+\sqrt{2k+2}}$. After rationalization we have ...
0
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0answers
36 views

increasing sequences.

Consider the sequence $\langle b_n\rangle$, where $b_n$ is given by $$b_n=\left \{\begin{array}{lr} 2\cot\frac {\pi}{n}, &\mbox{if $n\equiv 0\pmod 4$},\\ 2\csc\frac {\pi}{n}, &\mbox{if ...
4
votes
3answers
68 views

Find the sum of the $\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$

Let $0<p<1$,Find the sum $$\sum_{m=k}^{+\infty}\binom{m}{k}(1-p)^k\cdot p^{m-k}$$
2
votes
1answer
22 views

Limit of a sequence with a special property

There is sequence of real numbers,$(a_n)_{n\geq1}$, such that $\lim_{n\rightarrow\infty}a_n\sum_{k=1}^na_k^2=1$. How does it imply that $\sum_{k=1}^na_k^2\rightarrow\infty$ and ...
2
votes
2answers
50 views

Finiteness of the sums of reciprocals of positive solutions of $\tan x = x$ and $x = \tan \sqrt x$

Let $a_n$ be the sequence of positive solutions of the equation $\tan x=x$ and $b_n$ be a sequence of positive solutions of the equation $x=\tan \sqrt x$. Prove that $\sum \dfrac{1}{a_n}$ diverges ...
1
vote
0answers
16 views

Linear complexity of powers of a periodic sequences over finite fields

Let $\mathbf{a}^N = a_0 a_1\cdots a_{N-1}$ and let $\mathbf{a} = \mathbf{a}^N\mathbf{a}^N \cdots $ be an $N$-periodic sequence over the finite field $\mathbb{F}_q$ with $q$ elements, where $N \mid ...
5
votes
6answers
91 views

Finding the limit $\lim_{n\to \infty} \left({\frac{n+1}{n-2}}\right)^\sqrt n$

I have to find: $$\lim_{n\to \infty} \left({\frac{n+1}{n-2}}\right)^\sqrt n$$ But, to be honest, I haven't got a faintest idea how to even begin. Is there a way to evaluate this radical exponent?
3
votes
2answers
58 views

What is the minimal correction to the harmonic series such that it converges?

as you all hopefully know, the series $$ \sum_{k\ge 1}\frac{1}{k} $$ diverges. Now I know that you can add some logarithmic corrections, such that it converges: $$ \sum_{k\ge 1}\frac{1}{k\log(k)^2} $$ ...
0
votes
1answer
21 views

Find intervals where a series converges and uniformly converges

I'm not sure if I'm doing better. Here is the stuff. Consider the sequence of functions $$f_n(x) = nx \left(\frac{x}{n}\right)^n\text{sinc}^n\left(\frac{x}{n}\right)$$ and the series $$s(x) = ...
5
votes
1answer
114 views

How do you find the probability of A winning if the probability of getting a favourable outcome in the $r^{th}$ turn is a function of $r$?

Problem: Two players A and B are playing snake and ladder. A is at 99 and he needs 1 to win in rolling of a dice. However, he is always allowed to re-throw the dice if 6 appears. What is the ...
0
votes
1answer
57 views

If all elements of a sequence $x_n$ are in a set $M$ then $\lim_{n \rightarrow \infty} x_n$ is in $M$ as well? Why?

I'm wondering if this is true, for example if $M:=[1,3)$ then $\sup(M)=3$ but 3 is not in $M$.
5
votes
5answers
161 views

Find the value of $\sum_{n=0}^\infty\frac{1}{9n^2+9n+2}$

I was doing some problems in algebraic number theory and this series came up $$\sum_{n=0}^\infty\frac{1}{9n^2+9n+2}.$$ So, I would like to know the value of this series. However, I don't want a full ...
0
votes
2answers
25 views

Find the radius of convergence of this series

I used Dalamber's criteria, but when I solve the limit I find that it goes to infinity, which looks wrong. I think I might have done something wrong while simplifying the expression, but I don't quite ...
1
vote
2answers
21 views

Boundedness leading to pointwise convergence implying uniform convergence?

Consider a sequence of functions $\{f_n\}$ on some closed interval $I \subset \mathbb{R}$. Let's assume that $f_n$ is bounded on $I$ by $M \in \mathbb{R}$ for each $n \in \mathbb{N}$. If $\{f_n\}$ ...
0
votes
1answer
20 views

Find where $\sum _{n=0} ^\infty a_n(z+3-i)^n$ converges or diverges

If the power series $\sum _{n=0} ^\infty a_n(z+3-i)^n$ converges at $5i$ and diverges at $-3i$ then the power series: converges at $-2+3i$ and diverges at $2-3i$ converges at $2-3i$ and diverges at ...
4
votes
5answers
109 views

Calculate $\lim_{n\to\infty}(\sqrt{n^2+n}-n)$. [duplicate]

Introduction: An exercise from "Principles of mathematical Analysis, third edition" by Rudin, page 78. Exercise: Calculate $\lim_{n\to\infty}(\sqrt{n^2+n}-n)$. Explanation: I have a hard ...