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

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0
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0answers
6 views

Decide whether the following series converges $\sum_{n=1}^{\infty}\dfrac{(\ln n)^2}{n^{3/2}}$

Looking for a neat and smart way to solve this. I am having a tough time with this
0
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1answer
39 views

Convergence of $\sum_{n=2}^\infty n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt n}\right)$

I am trying to test the convergence of the series $$\sum_{n=2}^\infty n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt n}\right)$$ You can find this series in exercise 8.15 (l) - Mathematical Analysis 2nd ...
2
votes
1answer
27 views

Convergence of a Series involving Cos and Log

Does the following series converge? $$\sum_{k=1}^{\infty}\int_0^{\pi}\int_0^{\pi}\cos(2k(x-y))\log\big(\sin|\frac{x-y}{2}|\big)\,dx\,dy$$
0
votes
2answers
41 views

How to show that $\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$

Let $x_n$ be a sequence of reals. Show that $$\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$$ Since the weak convergence is equivalent to pointwise convergence of characteristic functions ...
1
vote
0answers
41 views

Particles reproduce after a delay, what is the population after $m$ months?

I have been working on the following puzzle: A seed particle produces 'r' particles/month. Each of these particles must wait at least 's' months before producing 'r' particles/month. ...
1
vote
1answer
37 views

Does $f_n(a_n)\to f(a)$ hold?

Say, we have $f, f_n \in C^0(\mathbb R, \mathbb C)$ such that $f_n \xrightarrow{\text{uniform}}f$ and a sequence of reals $a_n \to a$. Does it then hold that $f_n(a_n)\to f(a)$? I couldn't think of ...
0
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0answers
15 views

add 5 nos such that ans is 30 from following nos and nos can be repeated [duplicate]

use five nos from the following: 1,3,5,7,9,11,13,15 such that by adding them result is equal to 30 ++++_=30 nos can repeat themselves according to the question we ...
1
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2answers
51 views

A infinite sum with harmonic serie

Proof or disproof the folowing statment $$\sum_{n=1}^{+\infty}\frac{2n+1}{(n^2+n)^2}H_n=\sum_{n=1}^{+\infty}\frac{1}{n^3}$$ Where $\displaystyle H_n=\sum_{k=1}^{n}\frac{1}{k}$
11
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5answers
195 views

“Proof” that $1-1+1-1+\cdots=\frac{1}{2}$ and related conclusion that $\zeta(2)=\frac{\pi^2}{6}.$

Sorry if this has been posted before. Can somebody please tell me whether this result is correct, and give explanation as to why or why not? I'm not good at the formal side of maths. Start here: ...
1
vote
1answer
12 views

Series of points in a bounded sector of a complex half-plane

The question is: consider an infinite sequence of points which lie in a bounded sector of the complex plane, whose angular width is strictly less than pi (that is, it's an open sector of a ...
-1
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3answers
71 views

Sum of alternating series [on hold]

Looking to find the sum of the following series: $$\sum^\infty_{n=1}\frac{(-1)^n}{(2n+1)3^n}$$ It converges due to the Alternating Series Test.
0
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0answers
49 views

Simple sequence [on hold]

This is a puzzle I came across recently. I have n baskets. When we put a certain number of stones(x) into it the number x is doubled. I start with x stones. How do I distribute the stones across ...
2
votes
3answers
72 views

Sum of a geometric series whose common ratio might be 1

I want to sum a finite series with a computer but I need to allow the common ratio to be 1 or very close to 1. That means I can't use $$ \ S_n = \frac{1 - r^n}{1-r}\ $$ It needs to work reliably even ...
0
votes
2answers
17 views

Convergence of a sequence in absolut value.

I need to prove this: If $a_{n}$ converges to $A$, then $|a_{n}|$ converges to $|A|$. And I have this: $a_{n} \rightarrow A$ then, given $\epsilon>0$ there exists $N \in J$ such that ...
3
votes
1answer
48 views

Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$ 1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}} $$ More generally, are there general closed-form expressions for all continued ...
2
votes
2answers
35 views

formula for infinite sum of a geometric series with increasing term

I'm looking for the Expectation of the discrete random variable X, E[X], with pmf: $$p(x)=(\frac 16)^{x+1}, x=0,1,2,3...$$ so what I tried is as follows... $$E[X]= \sum_{0}^\infty xp(x) =$$ so then ...
-2
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1answer
49 views

Next term of this sequence [on hold]

I was doing some exercise on sequences and I'm stuck at this: 18,36,45,72,? The available answers are 19-63-46-87-29
2
votes
1answer
52 views

The meaning of infinite series $\sum_{i=0}^\infty 2^{-i}$, its relation to partial sums and Cantor's diagonal argument

Let's define $S(n)$ as $S(n) = \sum_{i=0}^n 2^{-i}$. Obviously, $\lim_{n \to \infty} S(n) = 2$ and also $\forall n \in \mathbb{N}, S(n)<2$. Now my questions are about $Q = \sum_{i=0}^\infty ...
2
votes
4answers
125 views

Addition of fractions repetition and convergence

Is this a new mathematical concept? $$ \frac{1}{n} + \frac{1}{n^2} + \frac{1}{n^3} \cdots = \frac{1}{n-1} $$ If it isn't then what is this called? I haven't been able to find anything like this ...
2
votes
1answer
53 views

Telescoping property: $\sum_{k \geq 0}\frac{1}{(4k+1)(4k+3)}$

I need to calculate the sum $$\sum_{k \geq 0}\frac{1}{(4k+1)(4k+3)}$$ I've made some attempts to transform this in a summation that I could apply the telescopic property, but I didnt have any ...
1
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2answers
39 views

Basic sequence question

Let $\{x_n\}$ be a sequence of real numbers. If $x_n\geq 0$ $\forall$ $n\in \mathbb{N}$, show that $\displaystyle x=\lim_{n\rightarrow \infty} x_n \geq 0$. I know that this is quite an easy problem ...
2
votes
1answer
54 views

How prove there exist postive integer $n$ such $x_{n}>y_{n}$

let two positive sequence $\begin{cases} x_{n+2}=x_{n}+x^2_{n+1}\\ y_{n+2}=y^2_{n}+y_{n+1} \end{cases}$ and $x_{1}>1,y_{1}>1,x_{2}>1,y_{2}>1$ show that: there exists $n$, such ...
4
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0answers
38 views

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

I'm trying to find a closed form of this sum: $$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$ WolframAlpha gives a large expressions containing multiple ...
0
votes
1answer
15 views

How to determine a general arithmetic sequence formula for two intersecting trig function

I have equations out of two trigonometric functions. For example $\cos(4\alpha$) = -$\sin(5\alpha)$ $\tan(0.5\alpha$) = 2 $\sin(\alpha)$ How can I determine a general arithmetic sequence formula ...
1
vote
1answer
39 views

Question about limit points in relation with continuity and functional limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have the feeling that the author is being careless about limit points in his theorems or I am not understanding ...
6
votes
2answers
113 views

Convergence of the series $\sum_{n=3}^\infty \frac{1}{(\log\log n)^{\log\log n}}$

I am trying to test the convergence of this series from exercise 8.15(j) in Mathematical Analysis by Apostol: $$\sum_{n=3}^\infty \frac{1}{(\log\log n)^{\log\log n}}$$ I tried every kind of test. I ...
0
votes
1answer
23 views

O Notation and taylor series

Wolframalpha tells me that the Taylor series of the exponential function is $1 + x + \frac{x^2}{2}+ O(x^3).$ Taylor series I just don't get this big O there, shouldn't this be a small o?
2
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4answers
55 views

general of partial sum of sequence

I am trying to find the limit of an infinite series given as $$\sum\frac{1}{n^2-1}.$$ I came across the following general term of the sequence of partial sums ...
4
votes
3answers
56 views

Evaluate $ \lim n \left[ 1-\frac{(n+1)^n}{en^n}\right] $

Evaluate the following limit of sequence $$\lim_{n\to +\infty} n \left[ 1-\frac{(n+1)^n}{en^n}\right] $$ I've transformed it in a 0/0 inequality and tried to apply L'Hospital one time, but the ...
5
votes
1answer
85 views

Find the sum, if exists $\sum\limits_{n=1}^{\infty} \frac{(2n)!}{2^{2n}(n!)^2(n+1)}$

$$ \sum\limits_{n=1}^{\infty}\dfrac{(2n)!}{2^{2n}(n!)^2(n+1)} $$ By comparison test this series converges. Any nice way to work the sum? I see that this can be written as: $$ ...
-3
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2answers
37 views

How to Prove that log(n)/n is not Cauchy? [on hold]

How can I prove that log(n)/n is Cauchy Sequence?
-2
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2answers
67 views

Judge the convergence of $\sum_{n=0}^\infty 1/\sqrt{n}$

How to judge the convergence of the sequence? $$\sum_{n=0}^\infty\frac{1}{\sqrt{n}}$$ Context I know two methods to judge whether a series converes: one is to calculate $\lim \frac{u_{n+1}}{u_{n}}$, ...
0
votes
4answers
31 views

Given formula to calculate sum of first n terms of a sequence, show that the sequence is geometric

Is there anything wrong with the following method to show that the sequence is geometric? It seems wrong because it uses the generic formula for sum of first n terms of a geometric progression ...
1
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1answer
35 views

In the definition of sequences diverging to infinity, why must the constants be positive?

We are given the following definitions A sequence $(a_n)$ diverges to $\infty$ if for each $ M \in \mathbb{R}^+ \exists N_M \in \mathbb{N} \ \text{such that } \\ a_n > M \ \forall n \geq N_M$ ...
-8
votes
1answer
49 views

Find next term of the sequence [on hold]

Find next term of the sequence $$4,8,61,23,46,821,652, ...$$ (Don't tell me that next term can be any number. Just find appropriate pattern, that can be described in a few words/formulas. Shorter ...
6
votes
3answers
139 views

Closed-forms of infinite series with factorial in the denominator

How to evaluate the closed-forms of series \begin{equation} 1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, ...
3
votes
6answers
128 views

Limit of sequences: $\lim \frac{(2n)!}{(n!)^2} $

Verify if the sequence $$\frac{(2n)!}{(n!)^2}$$ converges. My attempt: $$\frac{(2n)!}{(n!)^2} = \frac{(2n)(2n-1)...(n+1)}{n.(n-1)...1} \geq \frac{(n+1)^n}{n!} $$ Maybe it is easier to show that ...
1
vote
1answer
36 views

Proving that the upper and lower Riemann sums converge to the integral

This is a question from Tom M. Apostol's Calculus, Volume 1 (Exercise 10.4): $f$ is monotonically increasing and bounded on $[0,1]$. Define the sequences $\{s_n\}$ and $\{t_n\}$ as follows: ...
0
votes
2answers
56 views

Finding a formula for a pattern

I have this pattern which is an infinite sequence (I have placed commas so it's easy to see the pattern)... $1 ,1 2, 1 2 3, 1 2 3 4, 1 2 3 4 5 ...$ Is there any formula governing this sequence, ie, ...
2
votes
3answers
45 views

Proving that $\sum \frac{n^{n+1/n}}{(n+1/n)^n}$ diverges

Show that the series $$\sum \frac{n^{n+1/n}}{(n+1/n)^n}$$ diverges The ratio test is inconclusive and this limit is not easy to calculate. So I've tried the comparison test without success.
1
vote
1answer
25 views

$\sum r^n |\sin(nx)|$ convergence

Verify if the series $$\sum r^n |\sin(nx)|,\qquad r>0$$ Converges or diverges I've tried some comparisons with known series and the convergence tests, but didn't work. I think we ...
0
votes
0answers
41 views

convergence of a sequence of cauchy

I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric. There will be a metric space $X$ such that If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not ...
0
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0answers
36 views

necessary condition on the sequence $\{a_n\}$ and $\{b_n\}$ such that $\sum a_n b_n$ converges [on hold]

Is there any necessary condition on the sequence $\{a_n\}$ and $\{b_n\}$ such that $\sum a_n b_n$ converges ?
7
votes
1answer
118 views

Sum over all non-evil numbers

I'm working on the following contest math problem: Define an evil number to be any positive integer that contains the digit $9$. Show that $$ \sum_{x} \frac{1}{x} < 80 $$ where the ...
0
votes
0answers
26 views

Why does Cauchy's Root Test for convergence of infinite series require $\limsup$?

I'm confused about the reasoning behind Cauchy's root test for convergence of infinite series. It states that for any series $\{a_n\}$, if $C = \limsup_{n\rightarrow\infty}{\sqrt[n]{|a_n|}} < 1$, ...
2
votes
1answer
115 views

What is the infinite sum of $a^{b^x}$?

What would $$\sum^{\infty}_{n=0}(1/2)^{4^n}$$ be and how to determine it? EDIT: Apologies. I can see this converges by the ratio test. My issue is working out its sum, more for fun really. It ...
1
vote
1answer
19 views

In an irreducible, aperiodic, null-recurrent Markov chain holds $\sum_n p_{ij}^{(n)} = \infty$

My lecture notes state the following theorem: Theorem 2. Let $(X_n)$ be an irreducible, aperiodic, null-recurrent Markov chain. Then $$\forall i,j \in S : p_{ij}^{(n)}\to 0 \text{ and } \sum_n ...
3
votes
2answers
35 views

Sequence Convergence using bounding sequences

Consider the sequence $(a_n)$ with $a_n = F_{n+1}/F_n$ for $n \in \Bbb N$, where $F_n$ are the Fibonacci numbers. Show that this sequence converges to $\phi =(\sqrt{5}+1)/2$. Can someone help ...
4
votes
2answers
95 views

Challenging Infinite summation involving the zeta function [duplicate]

Evaluate: $$\large\sum_{k=1}^{\infty}\left(\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}\right)^2$$ MY ATTEMPT: Recognizing that $\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}$ can be written as $ ...
3
votes
5answers
81 views

Finite sequences of prime numbers

There is a lot of prime sequences: prime numbers in a special form. For example Mersenne primes are primes of the the form $2^n-1$, or Pythagorean prime are primes of the form $4n+1$. Even primes are ...