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

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21
votes
2answers
507 views

Would this solution of the limit of the sequence be correct?

Let's suppose that I have the sequence $a_n = \frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2} + \ldots + \frac{n}{n^2}, n \in \mathbb{N}$. And I have to find the limit of the sequence as $n \rightarrow ...
0
votes
1answer
52 views

Sequence $b_{n+1}=\frac{3-b_n^2}{2}$ - problem

Can anyone please help me with this because I honestly don't know how to start. Prove that sequence $b_n$ has limit and find it. Show that sequence $b_n$ is not monotonic and show that $b_n$ is the ...
3
votes
1answer
42 views

Does the series $\sum_{k=1}^n a^k \left(n-k\right),\ 0<a<1$, converge?

I have been trying to prove the convergence of the series $$\sum_{k=1}^n a^k \left(n-k\right)$$ for $0<a<1$, as $n \to \infty$, but the presence of the $n$ in the summands confuses me. I ...
9
votes
0answers
89 views

Find value to the summation : $\sum_{n =1}^\infty \dfrac 1 {5^{n+1}-5^n+1}$

$$\sum_{n = 1}^\infty \dfrac 1 {5^{n+1}-5^n+1}$$ I can factorize denominator to $4\times5^n+1$ to confirm the series does not diverge, But how do I calculate its actual sum? The series is not a ...
1
vote
1answer
38 views

Proving a sequence is bounded

I want to prove that (4n-1)/4n is bounded. My work: $$\begin{align} &\frac{4n-1}{4n}\\ = &\frac{\frac{4n}n - \frac 1n}{\frac {4n}n}\\ =&\frac{4 - 1/n}4\\ =&1 - \frac 1{4n}\\ ...
0
votes
0answers
26 views

Expand a $\arctan(x)$ function [duplicate]

I want to expand a function $\arctan(x)$ as a polynomial form. I know that I can use Taylor expansion in the case of x <1. But in my case, the x can be pretty large. Is there any way to expand or ...
0
votes
2answers
18 views

Infinite convergent sum with central binomial coefficient over k

Given the following sum: $$0.5\cdot\sum\limits_{k=0}^\infty \frac{1}{k+1}\binom{2k}{k}\cdot(0.25)^{k}$$ I know that the sum is supposed to converge to $1$. How would I go about evaluating it to get ...
0
votes
1answer
28 views

Which definition of convergence of subsequence is correct

Suppose that $(x_n)$ is a convergent sequence on a metric space $(M,d)$ with limit $x \in (M,d)$ Let $(x_{n_k})$ be the sub-sequence of the sequence $(x_n)$ Then is it more appropriate to write 1) ...
1
vote
2answers
50 views

Is there any way to differentiate such function?

Let $S$ be a set. If I had a bijection $f$ mapping each element $n\in \mathbb{N}$ to an element $s \in S$ such that: $$s = f(n) = \sum^{n}_{k=1} {1\over k}$$ Is the function differentiable in ...
0
votes
1answer
71 views

What is the pattern here: 1,2,5,10,13,26,39…

At first the pattern starts out ×2, +3, ×2, +3, ×2, but then jumps to +13? Can't see what the full pattern is. Any ideas?
1
vote
0answers
17 views

When the supremum of a real sequence is finite and not attained, it coincides with the limsup

I'm having a bit of a problem with an exercise I have to make. In the exercise we are given the sequence $(s_n)$, which is a sequence of reals. Furthermore, we are given that $m=\sup\{s_n|n \geq ...
0
votes
1answer
23 views

Radius of convergence of $\sum_{k=1}^\infty\frac{ln(1+\frac{1}{k})}{k}x^k$, I don't understand the solution

The exercise is to calculate the radius of convergence of $\sum_{k=1}^\infty\frac{ln(1+\frac{1}{k})}{k}x^k$. The solution of the book is the following: Because of ...
1
vote
2answers
60 views

If $(a_n)$ is decreasing and $a_{n+1}+a_n\to0$ then $a_n\to0$

Suppose that $(a_n)$ is a decreasing sequence of numbers such that $\displaystyle{\lim_{n\rightarrow +\infty} (a_{n+1}+a_n)=0}$. I want to show that $\displaystyle{\lim_{n\rightarrow +\infty} ...
0
votes
2answers
38 views

Proof by induction for recursive sequence with no explicit formula.

The problem I am trying to solve is: "show that the sequence defined by $a_1=6$ and $a_{n+1}=\sqrt{6+a_n}$ for $n\ge 1$ is convergent, and find the limit." So I know that I need to use proof by ...
0
votes
0answers
23 views

How to make inductive step for a Fibonacci proof [duplicate]

I have to prove $F^2_{n−1} = F^2_n + F^2_{n−1}$ for any $n >=1$ by induction (for the Fibonacci sequence). For the basis step, I have: $n = 1; $ $F_{(1)-1} = F^2_{(1)} + F^2_{(1)-1} ->$ $ ...
0
votes
1answer
28 views

Manipulating a power series expansion

By manipulating the power series expansion $\frac{1}{1-x} = \sum_{k=0}^\infty x^k, |x|<1$ Find a closed formula for $f(x)=\sum_{k=1}^\infty kx^k$ Not really sure how to go about this at all; I've ...
2
votes
0answers
25 views

a limit of recurrencive sequence with sine

consider the sequence $s_1=\sin 1$, $s_2=\sin\sin 1$, $s_3=\sin\sin\sin 1$ etc., the recurrence relation is $s_0=1$, $s_{n+1}=\sin s_n$. this sequence is bounded and decreasing, so convergent and ...
0
votes
0answers
10 views

Sufficient conditions differentiability in quadratic mean

I'm trying to show lemma 7.6 in van der Vaart "Asymptotic Theory" on the sufficient conditions for differentiability in quadratic mean of a probability density function but I have some doubts when it ...
0
votes
1answer
35 views

If $S_n$ is non-decreasing then $S_m \lt S_n$ whenever $m \lt n$

I am stuck on how to go about proving this: So I will list some of the facts of what I know at this point and maybe someone could push me in the right direction. Since $S_n$ is non-decreasing that ...
2
votes
5answers
75 views

Limit of the sequence defined by a recurrence

Given a recurrence formula for an arithmetic sequence, $$U_{n} = \frac{1}{2+U_{n-1}}$$ Show that$$\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+ ...}}}} = (SomeGivenValue)$$ How can we solve questions ...
9
votes
2answers
94 views

For any $n^2+1$ closed intervals of $\mathbb R$, prove that $n+1$ of the intervals share a point or $n+1$ of the intervals are disjoint

Stuck on a question from 'Introduction to Combinatorics by Martin J. Erickson'. Q: For any $n^2+1$ closed intervals of $\mathbb R$, prove that $n+1$ of the intervals share a point or $n+1$ of the ...
1
vote
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
1answer
63 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
votes
0answers
40 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
votes
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
vote
0answers
23 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
votes
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
votes
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
vote
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
votes
1answer
26 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
votes
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
votes
4answers
73 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
vote
2answers
40 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
13 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 ...