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

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1answer
51 views

Show that $\lim_{n \to +\infty} \frac{\sum^n x_i}{\sum^n y_i}=a.$ [duplicate]

Let $\{y_n\}$ a sequence so that $\sum y_i=+\infty$ and $$y_n>0, \forall n \in \mathbb{N}.$$ Show that if $$\lim_{n \to +\infty} \frac{x_n}{y_n}=a$$ then $$\lim_{n \to +\infty} ...
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1answer
73 views

If $a_n$ is prime then $n$ is prime too

Given sequence $(a_n)$ : $a_1=1, a_2=4, a_3=15, a_n=15a_{n-2}-4a_{n-3}$. Prove that if $a_n$ is prime then $n$ is prime too. It is easy to prove that $a_n=4a_{n-1}-a_{n-2}$ and ...
4
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1answer
123 views

Limit of this recursive sequence: $x_{n+1}=\bigl(1-\frac{1}{2n}\bigr)x_{n}+\frac{1}{2n}x_{n-1}.$

Consider the following sequence : $x_{0}=a$ , $x_{1}=b$ , $x_{n+1}=\bigl(1-\frac{1}{2n}\bigr)x_{n}+\frac{1}{2n}x_{n-1}.$ Find $\lim_{n\to \infty}x_{n}.$ I calculate $x_{2}$ , $x_{3}$ ,$x_{4}$ ,but ...
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2answers
54 views

Convergence and limits of recusive sequences.

I want to ask a question about recursive sequences. They have been pretty easy to handle for me, if you have one variable in it. To give you an example, if you have a sequence like: $x_o = 1, ...
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0answers
6 views

How the union of a bound series of integers converges to all integers for cases of all orders.

For the series $S_n = \{-n, \cdots, n \}^d$ I would like to show the union of all such sets converge to $\mathbb{Z}^d$ as $n \rightarrow \infty$. That is to said, how can I prove: $$\bigcup_{n \geq ...
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1answer
21 views

Alternating Series Convergence Area

I am asked to find for which values my series will converge. $$\sum^{\infty}_{n=0}\frac{x^{5n}}{(4+(-1)^n)^{3n}}$$ after root test I think I can write : when n is even : $$|\frac{x^5}{5^3}| < ...
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1answer
55 views

Can we simplify this sum?

Let $r>4$ and $n>1$ be positive integers. Can we simplify this sum: $$S=\sum_{m=1}^{n}\frac{2m}{r^{m^2}}$$ I have no idea to start.
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1answer
23 views

Residue of $g(z)$ at z=0 simple pole

Find the residue of: $$g(z) = \frac{\psi(-z)}{z(z+1)^2} \space \text{at} \space z = 0$$ My Attempt: Because $z=0$ is a simple pole, I thought of using the definition. $$\mathrm{Res} \space _{z=0} ...
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1answer
30 views

My proof of: Given an adherent point, some sequence converges to it.

(I have two specific questions.) Is my proof correct? Are style, wording, and punctuation alright? $\textbf{Definition 1.}$ Let $X \subseteq \mathbb{R}$. Let $x \in \mathbb{R}$. The point $x$ ...
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2answers
33 views

Write a sequence that is a geometric and arithmetic progression at the same time.

I thought to write this system $$\begin{cases} y_n = y_1 \cdot q^{n-1} \\ y_n = y_1 + (n+1)d \end{cases}$$ How do I solve it?
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0answers
22 views

Residue of a rational function

In this answer by Jack D'Aurizio, which is fantastic, I do understand that: If $f(z) = (\psi(-z) + \gamma)^2$ where $\psi(-z)$ is digamma, and $H_n$ (following) is harmonic number: $$\mathrm{Res} ...
2
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2answers
72 views

If $\lim\limits_{n\to +\infty}x_n=+\infty$ then $\left(\frac{x_n}{x_{n+1}}\right)_{n\in\mathbb{N}}$ converges

As in the title, in an exercise (Elementary Real Analysis by Thomson and Bruckner p.38), we have to prove that if $\lim\limits_{n\to +\infty}x_n=+\infty$ then ...
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2answers
56 views

$\lim_{{n}\to{\infty}} (n+1)!(e-1-{1\over2}-…-{1\over n!})= ?$

How can I solve this problem: Find the limit of the following sequence: $$\lim_{{n}\to{\infty}} (n+1)!(e-1-{1\over2}-...-{1\over n!})= ?$$ How to solve this using Cesaro-Stolz ? The numerator and ...
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1answer
76 views

About the solution of a difference equation

Let $r>4$ be a positive integer. Let us consider this difference equation: $$u_{n+1}=(1+r²ⁿ⁺¹)u_{n}-r²ⁿ⁻¹u_{n-1}+2$$ I want to find a closed form, bu I am not able to find the good idea.
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4answers
158 views

$\sum _{n=1}^{\infty } \frac{1}{n (n+1) (n+2)}$ Understand the representation

$$\sum _{n=1}^{\infty } \frac{1}{n (n+1) (n+2)}$$=$$\frac{1}{2} \left(-\frac{2}{n+1}+\frac{1}{n+2}+\frac{1}{n}\right)$$ $$s_n=\frac{1}{2} ...
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1answer
31 views

Can someone check my answers to this problem?

This is from Discrete Mathematics and its Applications Definition of recurrence relation from book From my understanding, compounded annually means that every year(annually) the account will ...
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2answers
88 views

Delta-Epsilon proof of $\lim_{n\to \infty} r^n = 0$ [closed]

Hey I've been having some trouble figuring out this problem. Not sure what to do with the restriction on $r$, or really how to go about writing it formally. Our professor assigned this as a problem ...
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1answer
440 views

Double sum and zeta function

This is a personal research that came to an end , since the results were not those which were being anticipated. I was unable to come up with a solution therefore I post the topic here: Prove (it ...
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2answers
204 views

Adding two convergent series

If $\sum_{n=1}^{\infty} a_n$ is finite and $\sum_{n=1}^{\infty} b_n$ is also finite, why is it that you can add the two series term by term and get the sum of the two series? Surely this is ...
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1answer
32 views

What are the necessary conditions for the Stolz-Cesàro theorem?

What are the necessary conditions for Stolz-Cesàro theorem? Let there be two sequences: $(x_n)$ and $(y_n)$. Is it possible to use this theorem if $(y_n)$ converges to zero?
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2answers
55 views

how interpret this partition identity?

use the symbol $P(N)$ to denote the set of all partitions of a positive integer $N$ and denote by $P_k$ the number of occurrences of $k$ in the partition $P \in P(N)$, so that $$ N = \sum kP_k $$ by ...
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1answer
37 views

Series ends with arithmetic series

Let us take series $a^2$. Where $a = 1, 2, 3, …$ The entire series of $a^2$ is look like: $A = 1, 4, 9, 16, …$ First step: The difference between every two terms of $A$ is: $B$ = $3, 5, 7, …$ ...
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1answer
69 views

Convergence of a sequence with assumption that exponential subsequences converge?

Problem One of my best friends asked me to think about the following problem: Suppose a sequence $\{a_n\}_{n=1}^\infty$ satisfies $\lim_{n\to\infty}a_{\lfloor\alpha^n\rfloor}=0$ for each ...
4
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1answer
43 views

Metric $p := p(x,y)= \min(|x-y|, 1- |x-y|)$ $x,y \in [0,1)^2$. Prove metric space is compact.

Help! I know that $X$ is Compact if every sequence in $X$ has a subsequence converging to a point in $X$. Also we have that $X$ is a bounded infinite subset in the real numbers. I think it's quite ...
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1answer
94 views

Find the Limit of the given sequence x_n = $(1 - 1/3 )^2$ $(1 - 1/6)^2$ $(1 - 1/10)^2$…$(1 - 2/n(n+1))^2$, n>=2

$x_n = \left(1 - \dfrac13 \right)^2\left(1 - \dfrac16\right)^2\left(1 - \dfrac{1}{10}\right)^2\cdots\left(1 - \dfrac{2}{n(n+1)}\right)^2, n\ge2$ Then find lim of (x_n) as n tends to infinity. I have ...
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3answers
83 views

Value and nature of $\sum_{n=1}^\infty(-1)^{n+1} \cdot \frac {1}{3^n}$

$$\sum_{n=1}^\infty(-1)^{n+1} \cdot \frac {1}{3^n}$$ Find the sum and determine it's nature. I have tried to find the limit of partial sums, but I could not do anything. There is another bunch of ...
2
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1answer
89 views

finding a harmonic sum using residues/complex analysis

Evaluate: $$S = \sum_{n=1}^{\infty} \frac{H_n}{n^2}$$ Using complex analysis. I just needs hints, I have no attempts, but I believe is has to do with residues.
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4answers
80 views

Sequence limit and monotony of $a_{n+1}=\sqrt{4a_n+3},a_1=5$

I am having difficulty proving whether the sequence is increasing or decreasing and finding its limit. Using derivative I find that it is increasing, but there seems to be another way. Computing ...
0
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1answer
26 views

Manipulating a summation expression for Future Value (Annuities)

I've been given the following expressions for regular payments with regular annual compounding (Annuities): (1) $$ F - (1 + r)F $$ (2) $$ F = d \sum^T _{t=1} (1+r)^{T-t}$$ where: ($F$) is a Future ...
2
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2answers
125 views

Proving that $\sum_{n=0}^{\infty }\frac{1}{(2n)!!}=\sqrt{e}$

Proving that $$\sum_{n=0}^{\infty }\frac{1}{(2n)!!}=\sqrt{e}$$ Firstly, I tried to check the value with the exponential function at $x=.5$ but I found its terms not equal to the series terms.Any help ...
0
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1answer
32 views

Uniform convergency of a given sequence of function

Consider the sequence of function $f_{n}(x)=n^{2}x\bigl(1-x^{2}\bigr)^{n}$. Is this sequence of function convergent uniformly in $[0,1]$ ? We find that $f$ converges point wise to the function ...
2
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1answer
72 views

Numerical convergence depending on summation order

I'm looking for an example of convergent series such that the numerical convergence depends on the order of summation? Or perhaps a series of positive terms where the partial sums value depend on the ...
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0answers
111 views

Simplification of recursive polynomials

Suppose I have known polynomials $p_i, i=0\ldots k-1$. I have the following horrific looking recursive equation: ...
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1answer
23 views

Tell if a series converges uniformly

Let $f(x) = \sum_{n=1}^\infty \frac{-2x}{(x^2+n^2)^2}$. Check if $f_n(x)$ converges to a continuous function. So I've seen a solution that uses the fact that if $f(x)$ converges uniformly and ...
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1answer
30 views

Prove convergence of this sequence $f(n)_{n \in \mathbb{N}}= \left(\frac{10+in}{n^2 + 2in}\right)^n$

I am having this sequence $f(n)_{n \in \mathbb{N}}= (\frac{10+in}{n^2 + 2in})^n$ Is this sequence bounded/ convergent? Thoughts: $lim_{n \to \infty}(\frac{10+in}{n^2 + 2in})=lim_{n \to ...
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1answer
37 views

Help with Baby Rudin Excercise 3.17. Partial converse to the fact that convergence impiies Cesaro summability. Warning: Pretty involved problem.

If $\{s_n\}$ is a complex sequence, define its arithmetic mean $\sigma_n$, by $$\sigma_n=\frac{s_0+s_1+\cdots+s_n}{n+1}.$$ Put $a_n=s_n-s_{n-1}$ for $n\ge 1$. Assume $M<\infty$, $|na_n|\le M$ for ...
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0answers
37 views

Finite sum of floor function. [duplicate]

Suppose I have the following finite sum; $$S_n=\sum_{k=1}^n{\left\lfloor{\frac{n}{k}}\right\rfloor}$$ I've never dealt with something like this before and was curious of a way to express it with a ...
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2answers
30 views

How to bound the tail of p-series

How can I asses $S_n = \sum_{j=n}^\infty\frac{1}{j^p}, p>1$ in terms of $n$, specifically can I get something like $$S_n = O(?)$$
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1answer
23 views

Understanding uniformly bounded sequence

The sequence $(a_i)_{i\in I}$ is uniformly bounded if there exists a number $M<\infty$ such that $|a_i|\leq M,\forall i\in I$. Why can't I say that the sequence is uniformly bounded when $|a_i| ...
2
votes
1answer
31 views

Is $\sum^{\infty}_{n=1}xe^{-n^{2}x}$ convergent on specified intervals?

Is $\sum^{\infty}_{n=1}xe^{-n^{2}x}, \ x \geq 0$ convergent uniformly on specified intervals?
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2answers
37 views

Convergence of $\sum^{\infty}_{n=1}\dfrac{1}{n^{2+x}}$

Does $$\sum^{\infty}_{n=1}\dfrac{1}{n^{2+x}}, \ x \geq 0$$ converge uniformly on specified intervals? I know that $\sum^{\infty}_{n=1}\dfrac{1}{n^{2}}$ converges when $|x|<1$. But what about ...
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2answers
28 views

Working out values of sequences using partial sums

I have the sequence $a_n =\sqrt{n+1}-\sqrt{n}$ and $b_n=\dfrac{1}{\sqrt{n}}$ let $s_n$=$a_1 + a_2 + a_3 + ... + a_n =\displaystyle \sum_{k=1}^n a_k$ and $t_n$=$b_1 + b_2 + b_3 + ... + b_n = ...
2
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2answers
72 views

Is there any result that could help me obtain a lower bound for this series?

The series has the structure $\sum_{k=1}^\infty p_ka_k$ and converges to some $c<0$. $\{a_k\}_{k=1}^\infty$ is known, is an increasing function of $k$ with $a<a_k<0, \forall k$, has limit ...
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2answers
53 views

Compute the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$

What would be the sum of the following series? $$\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$$ Thanks
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0answers
17 views

How to evaluate the following q-binomial series?

How to evaluate the following series $$\sum_{n=2}^{\infty}x^n\sum_{k=2}^{n-1}(a+k~b)\prod_{j=k}^{n-1}(1+c~q^j)$$
2
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1answer
36 views

How to evaluate the infinite series $\sum _{i=0}^{\infty } \rho ^i \prod _{j=1}^i \left(\frac{\alpha }{j}+1\right)$

Mathematica shows that the infinite series \begin{align}\sum _{i=0}^{\infty } \rho ^i \prod _{j=1}^i \left(\frac{\alpha }{j}+1\right)= -\frac{(1-\rho )^{-\alpha }}{\rho -1}. \end{align} How do I ...
1
vote
2answers
24 views

Proving convergence by convergence of 3 subsequences

I'm studying for my real analysis exam and came across this prompt: Let $\{a_n\}$ be a sequence of real numbers. If the subsequences $\{a_{2k}\}$, $\{a_{2k+1}\}$and $\{a_{3k}\}$ converge, prove that ...
2
votes
4answers
77 views

Proving that the series 1 + … + $1 / \sqrt{x}$ < $2 \sqrt{x}$

Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$ I am doing it by simple induction adding $1/\sqrt{x+1}$ to both sides, but I can't find a way to manipulate this expression and find ...
1
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2answers
63 views

Upper Bound for $|f^{n}(0)|$ given that $f$ is Analytic

Let $f(x)$ be an analytic function in some neighborhood of $x=0$. $f$ being analytic implies that its has a convergent Taylor series expansion about $x=0$. That is, there exists $R>0$ (radius of ...
2
votes
4answers
129 views

Prove that $p$ divides $F_{p-1}+F_{p+1}-1$ [duplicate]

Given the Fibonacci sequence $(F_n)$, defined by $F_0=0,F_1=1, F_{n+2}=F_{n+1}+F_n$, and $p$ an odd prime number, how to prove that $p$ divides $F_{p-1}+F_{p+1}-1$? Is induction a good idea here? ...