For questions about recurrence relations, convergence tests, and identifying sequences

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18
votes
9answers
3k views

Is the Look and Say Sequence a “proper” maths problem? [closed]

I once told my brother about the "Look and Say Sequence" (i.e. 1, 11, 21, 1211, 111221, ...). My brother then showed the sequence to his maths teacher at school and asked him to predict the next ...
2
votes
0answers
114 views

Summary of divergent series summation methods and relations between them?

There are a number of methods of assigning sums to series that do not necessarily converge, e.g. Cesàro summation, Abel summation, Ramanujan summation, etc. (There is also the trivial method of only ...
1
vote
2answers
681 views

Prove $x^n$ is not uniformly convergent

This question pertains to the sequence of functions $f_n(x)=x^n$ on the interval $[0,1]$. It can be shown this sequence of functions ${f_n}$ converges point-wise to the limit $f$ where $f$ is defined ...
2
votes
1answer
41 views

Is 'limit' synonymous with 'radius of convergence'?

I of course read the Wikipedia article, but it sounded like such an abstract idea, that I could interpret only as a limit. The term appeared in a recent lecture apparently out of nowhere (though I ...
4
votes
2answers
309 views

Test for convergence of the series $\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$

Could I have a hint for testing the convergence of the following series please? $$\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$$ Edit The integral test does not work because $\int_1^n\frac{1}{(\ln ...
2
votes
3answers
97 views

What would be the limit

What would be the limit of: $$\lim_{n \to \infty}\frac{1^2+3^2+...+(2n-1)^2}{2^2+4^2+...+(2n)^2}$$ Knowing that $2n-1$ is an arithmetic sequence the sum of it would be $\frac{n(a_{1}+a_{n})}{2}$, ...
0
votes
2answers
38 views

Absolute value of the difference between two sequences

Consider two sequences $(a_n)$ and $(b_n)$ both contained in $\mathbb{R}$ and assume that these two sequences satisfy $|a_n - b_n| \rightarrow 0$, then this does imply that both $a_n$ and $b_n$ are ...
2
votes
5answers
105 views

A convergence proof

I have to proof that $$\lim_{n\rightarrow\infty} \left(1+n^2\right)^{\frac{1}{n}}$$ converges and give its limit. What I did was the following: $$\lim_{n\rightarrow\infty} ...
0
votes
1answer
50 views

Prove $\limsup\{a_n\}\ge 1$

Let $\{a_n\}$, a bounded sequence such that $a_n>0 \forall n>0$. Let: $$\mathop {\lim }\limits_{n \to \infty } {a_{n + 1}}{a_n} = 1$$ Prove that $\displaystyle\limsup_{n\to\infty} a_n\ge ...
1
vote
0answers
28 views

Renormalization operator R

In a Robert Devaney's book ("an introduction to chaotic dynamical systems") is approached the quadratic map $F_\mu=\mu x (1-x)$. He introduce the renormalization operator R. R is a function of ...
0
votes
1answer
39 views

a Problem about Special Sequence

I am doing homework. And this question may related to #Proving statement about sequences. However, I want to consider the limit of it. Let we assume $a_1=a,b_1=b,b>a>0$. Then we consider the ...
1
vote
1answer
31 views

Multiplying of sequences

Let $(x_n)$ be a sequence of positive numbers such that $\lim\limits_{n\rightarrow \infty}x_n=0$. Prove that there exists a decreasing sequence of positive numbers $(y_n)$ such that series $\sum y_n$ ...
0
votes
1answer
93 views

a Problem about Sequence [duplicate]

Let $a_1$ be an integer. Then we assume $$ a_{n+1} = \begin{cases} 3a_n+1,&\text{$a_n$ is odd}\\ \frac{a_n}{2},&\text{$a_n$ is even} \end{cases} $$ Now we prove that for any ...
1
vote
2answers
99 views

Polynomials through successive differences

Let $h_0:\Bbb{N}\rightarrow\Bbb{N}$ be any function. Define recursively, for $m>0$, $$h_{r+1}(m)=h_r(m)-h_r(m-1).$$ Suppose that for some $k>0$ we have $h_k(m)\equiv d$ constant. Is this ...
0
votes
2answers
43 views

Help with Interchanging Limits

How can I evaluate: $$\lim \limits_{n\to \infty}\frac{1}{N}\left(\sum_{m=1}^\infty\frac{1}{m^2}\left|\frac{\sin(mN)}{\sin(m)}\right|\right)$$ I know that we have ...
-1
votes
1answer
23 views

Evaluate: $x_i\ge0; \max_{1\le i \le n , \sum_{i=1}^{n}{x_i=a}}x_1x_2…x_n$

Let a be a fixed positive real number.Evaluate: $$x_i\ge0; \max_{1\le i \le n , \sum_{i=1}^{n}{x_i=a}}x_1x_2...x_n$$ I guess it should be $x^n$ where $x=\frac{a}{n}$since this is true for $n=2$, I ...
0
votes
2answers
72 views

About the sequence recursively defined by $x_{n+1}=b\left(1-\frac{b}{4x_n}\right)$

I don't know how to handle this: Given $b > 0$, consider the following recursively defined sequence: $x_1=b$ $x_{n+1}=b\left(1-\frac{b}{4x_n}\right)$, for all $n \ge 1$. I've seen that the ...
2
votes
0answers
62 views

Help with Series Convergence

Can someone help me prove that the following series $S$ converges: $$S=\sum_{m=1}^\infty\frac{1}{m^2|\sin(m)|}$$ I would appreciate any help in constructing a simple proof.
6
votes
3answers
113 views

Calculate the limit of $nx_n^2$

Let a sequence, $\{x_n\}$ such that: $x_{n+1}=x_n-x_n^3$ and $0<x_1<1$. 1) Prove $\mathop {\lim }\limits_{n \to \infty } {x_n} = 0$ 2) Calculate $\mathop {\lim }\limits_{n \to \infty } ...
0
votes
1answer
43 views

Show a sequence converges (Probably using Cauchy's sequence Thm)

This demand is a part of a proof. It must be easy, I'm just failing showing it rigorously. Let: $\left| {x_{n+1} -x_n} \right| < {1\over 2^n}$ We want to prove it's a Cauchy's sequence: ...
0
votes
1answer
106 views

Arithmetic progression question

In an arithmetic progression there are $2n$ arguments. The sum of last $n$ arguments is three times greater than that of the first n arguments. It is also known that the last argument in the series is ...
1
vote
1answer
110 views

Convergence Test $\sum_{n=2}^\infty\frac{1}{n^p-n^q}$

To test the convergence of the series $\sum_{n=2}^\infty\frac{1}{n^p-n^q}$ I tried the limit comparison test. $0<q<p.$ My $a_n=n^p-n^q$ and my $b_n=\frac{1}{n^p-n^q}$. The limit comparison test ...
3
votes
1answer
92 views

Prove $\{x_n\}$ converges

Let $\{x_n\}$, a sequence such that: $\forall n \in \mathbb{N}: \left| {x_{n+2}-x_n} \right| < {1 \over 2^n}; \quad \lim_{n \to \infty } ({x_{n + 1}} - {x_n}) = 0$ I translated the above ...
5
votes
2answers
3k views

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4} $using Parseval's theorem (Fourier series)

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4} $using Parseval's theorem (Fourier series). I have , somehow, to find the sum of $\sum_{n=1}^\infty \frac{1}{n^4}$ using Parseval's theorem. I tried ...
1
vote
2answers
71 views

Prove that this sequence converge.

I am obliged to prove that this sequence: $\large {a_n=(1+\frac{1}{3})(1+\frac{1}{9})(1+\frac{1}{27})...(1+\frac{1}{3^n})}$ is convergent sequence. I mean I was thinking about this and I know that ...
1
vote
2answers
315 views

Adding sum of negative integers to the sum of positive integers - how would value look like?

I know that the sum of positive integers will diverge and that the sum of negative integers will diverge. I also know that "the sum of all integers" will likely be meaningless (it depends on the order ...
4
votes
3answers
58 views

Find the largest interval for which the following series is convergent at all points x in it

Find the largest interval for which the following series is convergent at all points x in it. \begin{equation} \sum_{n=1}^{\infty} \frac{2^n(3x-1)^n}{n}. \end{equation} Applying ratio test radius of ...
1
vote
2answers
82 views

Ariawan's Ratio: $\mathrm{A}$

Given $$\mathrm{A}=\lim_{N \to \infty}\left|\frac{\sum\limits_{\alpha=0}^N \frac{(1 + i\pi)^\alpha}{\alpha!}}{\sum\limits_{\alpha=0}^N \frac{1}{\alpha!}}\right|,$$ then determine the value of ...
0
votes
1answer
62 views

The series $a_n$ converges and $a_n>0$. Check if the series $\frac{\sin a_n}{a_n}$ converges.

I tried to prove that it does not converge by showing that the series $b_n=\frac{\sin a_n}{a_n}$ does not converge to $0$. But I am stuck with a limit of the kind $\frac{0}{0}$. Thanks for ...
0
votes
1answer
58 views

Combination of monotone and bounded sequences

If $a_n$ is a monotone and bounded sequence, prove that: $b_n=\frac{a_1+a_2+...+a_n}{n}$ is also monotone and bounded. What is the way to prove this?
0
votes
1answer
60 views

Application of binomial theorem to simplify

$a_n=\sqrt[n]{n}-1$ Use the binomial theorem to show that $n \ge \frac{n(n-1)}{2} a_n^2$ Please help with this problem - how do I apply the binomial theorem here? What do I choose for the k of n ...
1
vote
3answers
60 views

Convergence of two Cauchy Sequences

Suppose that $a_n$ and $b_n$ are Cauchy sequences, and that $a_n < b_n$ for all n. Prove that $\lim_{x \to \infty}a_n \le \lim_{x \to \infty}b_n$ for all n. What is the appropriate strategy to ...
0
votes
2answers
92 views

Cauchy sequence convergence

Suppose that $a_n$ and $b_n$ are Cauchy sequences, and that $a_n < b_n$ for all n. Prove that $\lim_{x \to \infty}a_n \le \lim_{x \to \infty}b_n$ for all n. Is it sufficient to say that we know ...
1
vote
4answers
105 views

Why are these two series identical?

Could someone please show/explain to me explicitly why this is true (from wiki): $S = 1 - 1 + 1 - 1 + 1 - 1 + \cdots$ The series can be rearranged as: $S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + \cdots)$ I ...
1
vote
1answer
77 views

Find a uniformly continuous function such that $a_{n+1}=f(a_n)$

$a_{n+1} = a_n - a_n^2$, $a_1 = 2/3$. for $n\ge1$ a) Show the series converges and find its limit. b) find a uniformly continuous $f:\mathbb{R}\rightarrow \mathbb{R}$ such that: ...
2
votes
1answer
69 views

$\sum_{1}^{\infty}a_{n}$ converges $\Rightarrow \sum_{1}^{\infty}\mathrm{Log}(1+a_{n})$ converges, $a_{n} \in \mathbb{C}$

Let $a_{n} \in \mathbb{C}$ ; is it true that convergence of $\sum_{1}^{\infty}a_{n}$ implies convergence of $\sum_{1}^{\infty}\mathrm{Log}(1+a_{n})$ ? Why ? Here $\mathrm{Log}$ is the the principal ...
0
votes
1answer
32 views

Given $k$ sequences: $(x_n^1),(x_n^2),…,(x_n^k)$ how can you build a sequence that these are its sub sequences?

Given $k$ sequences: $(x_n^1),(x_n^2),...,(x_n^k)$ how can you build a sequence that these are its sub sequences ? I notice that each of those sequences have a different limit but other than that ...
1
vote
1answer
121 views

Use monotonic property of subsequences to prove one sided limits at a point of a monotonic function

We were given a two parter homework question: Prove that if every subsequence of a sequence $x_n$ is converging to L then $x_n$ converges to L. Use what you proved above to prove that if $f(x)$ is a ...
2
votes
2answers
58 views

How find this $a^2_{n}-2b^2_{n}=?$ [closed]

let $\{a_{n}\},\{b_{n}\}$ such $a_{0}=b_{0}=1$ and $$a_{n}=a_{n-1}+2b_{n-1},b_{n}=a_{n-1}+b_{n-1},n=1,2,\cdots,n$$ show that $$a^2_{n}-2b^2_{n}=?$$
1
vote
5answers
245 views

Does $\frac12+\frac14+\frac18+\dots$ equal $1$? [duplicate]

Suppose I have something with length one unit. I divide it to two equal length $0.5$ unit and put left one in my left side and right one in my right side. I then do same for my right side and contact ...
2
votes
1answer
396 views

Test for convergence of $\sum_{n=2}^\infty(\ln n)^p$

What test do you suggest for testing the convergence of the series $\sum_{n=2}^\infty(\ln n)^p$ when $p<0$? I have (I hope I did it correctly) already tested the convergence for $p\ge0$ using the ...
3
votes
2answers
83 views

when is this statement is true?

Let $a_i>0$ be a sequence in $\mathbb{R}$. It's well known that: $\sum\limits_{i=0}^{n}a_i\to a \ $(as $n\to\infty)\Longrightarrow a_i\to0$ (as $n\to\infty$) My question is when is the following ...
1
vote
2answers
112 views

The sum of a telescoping series.

How can I proof that the sum of $$1- \frac12 + \frac13 - \frac14 +\frac15 -\, \dots$$ up to $\infty$ is $\ln(2)$? How can I calculate it using the telescoping method? Thank you
3
votes
3answers
184 views

Finding limit of a sequence in product form

\begin{equation} \prod_{n=2}^{\infty} \left (1-\frac{2}{n(n+1)} \right )^2 \end{equation} I need to find limit for the following product..answer is $\frac{1}{9}$. I have tried cancelling out but ...
3
votes
2answers
257 views

What consistent rules can we use to compute sums like 1 + 2 + 3 + …?

$ \newcommand{ifelse}[3]{ \left( \begin{cases} #1\text{ if }#2 \\ #3\text{ otherwise} \end{cases} \right) } $ A recent Numberphile video on 1+2+3+... has made this question ("Why?") popular again, as ...
0
votes
1answer
26 views

Pitt's theorem on automatic compactness of bounded operators between sequence spaces

Why is it called Pitt's theorem? I couldn't locate the origin of the statement.
1
vote
3answers
51 views

Under certain conditions, prove: If $\lim_{x\to\infty}f'(x) = 0$ and $\lim_{n\to\infty} x_n = \infty$ then $\lim_{x\to\infty}f(x_{n+1}) - f(x_n) = 0$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $\lim_{x\to\infty} f'(x) = 0$. Furthermore let $(x_n)_{n\in\mathbb{N}}$ be a sequence of real numbers with $\lim_{x\to\infty}x_n = ...
1
vote
1answer
99 views

What are the coefficients of these trigonometric sums?

I have two functions that I'm working on. The first is: $$ \begin{align} \cos x &= (\cos 1)^3 \cos(3-x) \\ &{}+ 3 (\cos 1)^2 (\sin 1) \sin(3-x) \\ &{}- 3 (\cos 1) (\sin 1)^2 \cos(3-x) \\ ...
1
vote
4answers
76 views

Given $(a_{n+1} - a_n) \rightarrow g$ show $\frac{a_n}{n} \rightarrow g$

So yea, basically this is the problem. $(a_{n+1} - a_n) \rightarrow g$ show $\frac{a_n}{n} \rightarrow g$ It looks like Cauchy's sequence, but I'm not sure. Can we say that if $(a_{n+1} - a_n) ...
3
votes
7answers
178 views

Proof of $\frac{1}{(1-r)^2}$ [duplicate]

I was given that $S=1+2r+3r^2+\cdots = \frac{1}{(1-r)^2}$ was an identity from my students, and I tried to prove directly using geometric series. I got stuck and looked around online only to be told ...