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

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3answers
53 views

Geometric sequence problem

Determine the value(s) of k, so that the positive numbers $\log_8(k-1)$, $3\log_8(k-1)$ and $6$ form a geometric sequence (in order given above).
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3answers
34 views

Arithmetic and geometric sequence

Which two numbers should be placed between -5 and 49 so that the first three numbers form an arithmetic sequence, whereas the last three numbers form a geometric sequence?
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2answers
68 views

Is the Taylor series of the $f: \mathbb{R} \to \mathbb{R}$ evaluated at $0$ converges pointwise (in the whole $\mathbb{R}$)?

I would like to solve this problem: Consider the $f: \mathbb{R} \to \mathbb{R}$ function which is $n$-times differentiable for any $n \in \mathbb{N}$. Is it true that the Taylor series of ...
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2answers
77 views

Prove $\lim \frac{3^n + 2^n}{5\cdot3^n + 7\cdot2^n} = \frac{1}{5}$

Prove using limit definition. $$\lim \frac{3^n + 2^n}{5\cdot 3^n + 7\cdot 2^n} = \frac{1}{5} $$ My try: $$\left| {\frac{3^n + 2^n}{5\cdot 3^n + 7\cdot 2^n} - \frac{1}{5}} \right| < \varepsilon ...
1
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1answer
20 views

Meaning Behind Mapping from a Compact Subset to Another Set

Suppose I tell you that a set, $A$, is compact and a subset of a metric space. This means that it is closed and bounded and that every sequence in set $A$ has a converging sub-sequence. Then I tell ...
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2answers
28 views

finding for which $c\in \mathbb{R}$ sequence converges

so i am trying to find for which $c\in\mathbb{R}$ this sequence converges: $a_{1}=c$ and $a_{n+1}=1+\frac{a_{n}^2}{4}$ So i got the basic idea how to do this. First i found the candidate for limit: ...
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3answers
41 views

Product of first $n$-th prime power integers $+ 1$

I was just playing with prime numbers and then I accidentally found this pattern. Let $p_1\cdot p_2\cdot p_3\cdots p_n$ is the product of first $n$-th prime power integers. Prove that: $p_1\cdot ...
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4answers
87 views

About the limit $\lim_{n\to +\infty}\frac{n^k}{n!}$ for a fixed $k\in\mathbb{N}$.

Given a natural number $k$ and some real number $\epsilon>0$, I have to prove that there exists a natural number $n$ such that $\frac{n^k}{n!}<\varepsilon$. I tried to develop for ...
2
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3answers
320 views

Which is the limit of the sequence $\sum_{i=1}^n \frac{\cos(i^2)}{n^2+i^2}$

I can't find the limit as n $\to$ infinity of the sequence: $$\frac{\cos(1)}{n^2 + 1} + \frac{\cos(4)}{n^2 + 4} + \dots + \frac{\cos(n^2)}{n^2 + n^2}$$ I tried to use the inequality $\cos(n^2) < ...
4
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0answers
77 views

Limit of $\frac{1-2+\cdots+(2n-1)-2n}{\sqrt{ (n^2+1)}+ \sqrt{ (n^2-1)}}$

Here is the limit to be calculated : $$\lim_{n\to\infty}\frac{1-2+\cdots+(2n-1)-2n}{\sqrt{ (n^2+1)}+ \sqrt{ (n^2-1)}}$$ The question provides no other information. Doubts What is the domain of ...
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1answer
34 views

Inverse sum representation of sine

The other day I was playing with functions of the form $$ f(x) = \frac{1}{\frac{1}{a_0(x-b_0)} + \frac{1}{a_1(x-b_1)} + \cdots + \frac{1}{a_n(x-b_n)}} $$ and I found particularly that $$ ...
2
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1answer
40 views

Use of restriction to disallow aberrant series

My question concerns restrictions on the exercise of normal algebraic rules. The most well known restriction is the prohibition on division by zero (PDZ). This is justified by various 'proofs' of ...
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0answers
46 views

Convergence Intervals

Consider the function $$f(x) = \frac{x}{1-x}$$ We know that for $x\in(0,1)$, $$f(x) = x\cdot\frac1{1-x} = \sum_{k=0}^\infty x^{k+1} = \sum_{k=0}^\infty x^{k} - 1$$ Now, notice that: $$\frac{x}{1-x} = ...
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2answers
64 views

Show that $\sum_{n=0}^\infty\left(\frac{1}{n+1} - \frac{1}{n+2}\right)= 1$ [closed]

Show that $$ \sum_{n=0}^\infty\left(\frac{1}{n+1} - \frac{1}{n+2}\right)= 1 $$
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1answer
41 views

Examine the uniform convergence of the series $\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$ if $x \in [0, \infty]$

Examine the uniform convergence of the series $$\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$$ if $x \in [0, \infty)$ Which series should I choose in Weierstrass M-test to show that is divergent? ...
3
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0answers
42 views

Summing the binomial pmf over $n$, part 2

After the great answers I got to this question, I tried summing a similar-looking series using the same strategies ($k \geq 0, \alpha > 1, p \in (0,1)$): $$ \sum_{n=k}^{\infty} {n \choose k} p^k ...
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2answers
74 views

What will be the 100th term of the series: [closed]

If $a_1 = 1$ and $a_{n+1}-3a_n+2=4n$ for every positive integer n, then find the value of $a_{100}$
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2answers
20 views

Variation of geometric, harmonic and arithmetic means in sequence.

A question I got on my test was - Let ${A}_{1}, {G}_{1}$ and ${H}_{1}$, denote the arithmetic, geometric and harmonic means of two distinct positive numbers. For $n\geq 2$, Let ${A}_{n-1}$ and ...
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1answer
81 views

Questions from an olympiad on number theory [closed]

The sum of the infinite series: $$ \frac{1}{2} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +....$$ I am able to find the general term ...
3
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0answers
33 views

How can I calaculate this complex series? [closed]

Let $A=\{a+bi|a,b\in \mathbb{Z}\}-\{0\}$ For what $n$ does $\sum_{z \in A} 1/z^n$ $(n\in \mathbb{N})$ converge? If it converges, how can I evaluate it? Thanks in advance.
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2answers
68 views

For $l^2 (\mathbb{N})$ is $(l^2, d_2)= (l^2, d)$ topologically, where $d$ forms the usual product topology on $\mathbb{R}^{\mathbb{N}}$?

If we define $d(x,y)= \max_{n \in \mathbb{N}}( \min \{2^{-n}, |x_n - y_n| \})$ then does this distance form the same topology on $l^2 ( \mathbb{N})$ (the set of all square-summable real sequences) as ...
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2answers
42 views

Infinite series of trigonometric ratios

The question is to compute: $$(1+\cos A)+2(1+\cos A)^2 + 3(1+\cos A)^3+\ldots = \sum_{k=1}^{\infty}k(1+\cos A)^k.$$ I tried by setting $1+\cos A=y$, then the serie becomes $$y+2y^2+3y^3+\ldots = ...
0
votes
1answer
39 views

For which values of $p$ does the serie $\sum\limits_{n=2}^\infty\frac{1}{n^p\ln(n)}$ converge?

For which values of $p$ does the serie $\sum\limits_{n=2}^\infty\frac{1}{n^p\ln(p)}$ converge? I'm trying to use the ratio test but I can't get a simple term in which use limit easily enough.
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0answers
24 views

Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure

Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure and $ f_n(x)=\dfrac{|\cos(x^{-2})|}{x^{1-1/n}} $ for $ x\in (0,1] $ Is there lebesgue integrable function $g$ ...
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3answers
25 views

$a_n - \frac{a_n - v}{s}$ becomes $v$

Given is following sequence: $a_{n+1} = a_n - \frac{a_n - v}{s}$ I found out that $\forall a_0, v, s \in \mathbb{R}, s>0: \lim\limits_{n \to \infty}a_n=v$ But I do not know why. I tried to ...
2
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2answers
52 views

Enumerating the positive rationals without repetition

From Makarov's Selected problems in real analysis Let $r_n$ be defined as follows\begin{cases} r_1=1 & \\ r_{2k}=1+r_k \\ r_{2k+1}=\frac{1}{r_{2k}} & \end{cases} ...
2
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1answer
45 views

A proof that $\frac{(2\phi)^n-(-1)^n}{\phi^{2n}-(-1)^n}\cdot\left(2^n-\phi^n\right)\cdot\sqrt5\in\mathbb Q$ for all $n\in\mathbb Z$

During computation of some series (with help of a CAS), at an intermediate step I encountered an expression, that after dropping non-essential parts looks like this:$$\mathcal ...
3
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0answers
30 views

Show the convergent of sequences and limit? [duplicate]

a)Show that the following sequence of functions is convergent. $f_{1}(x) = \sin(x)$ $f_{2}(x) = \sin(\sin(x))$ $f_{3}(x) = \sin(\sin(\sin(x)))$ ... $f_{n}(x) = \sin(\sin(\sin(...(\sin(x))))$ ...
2
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3answers
71 views

Summing the binomial pmf over $n$

I was trying to work out some bounds for a research problem when I came across the innocuous-looking sum: $$ \sum_{n=k}^{\infty} {n \choose k} p^k (1-p)^{n-k}, \quad k \in \mathbb{N}, \; p \in (0,1)$$ ...
28
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1answer
387 views

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

There is a known asymptotic expansion of harmonic numbers $H_n$ for $n\to\infty$: $$\begin{align}H_n&=\gamma+\ln n+\sum_{k=1}^\infty\left(-\frac{B_k}{k\cdot n^k}\right)\\ &=\gamma+\ln ...
0
votes
1answer
22 views

How many terms lies between

If a have a series from $n^2$ to $(2n)^2$, i.e. $n^2+(n+1)^{2} +...+ (2n)^2$, how many terms lie between $n^2$ and $(2n)^2$ ? Is it $n+1$ terms, or is it $n$ terms, and how to prove that.
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3answers
109 views

If $\lim_{n\to\infty} x_{n+1}-\frac12 x_n = 0$ then $x_n\to 0$

Let $x_n$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty} x_{n+1}-\frac12 x_n = 0$ Prove that $\lim_{n\to\infty} x_n = 0$ I have a proof if one assumes that $(x_n)$ ...
2
votes
3answers
60 views

Let $A_n=\frac{(n+1)+(n+2)+(n+3)+…+(n+n)}{n}$,$B_n=[(n+1)(n+2)(n+3)…(n+n)]^{1/n}$.If $\lim_{n\to\infty}\frac{A_n}{B_n}=\frac{ae}{b}$

For positive integers $n$,let $A_n=\frac{(n+1)+(n+2)+(n+3)+.....+(n+n)}{n}$,$B_n=[(n+1)(n+2)(n+3)....(n+n)]^{1/n}$.If $\lim_{n\to\infty}\frac{A_n}{B_n}=\frac{ae}{b}$ where $a,b\in \mathbb{N}$ and ...
3
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2answers
215 views

Where does this sequence $\sqrt{7}$,$\sqrt{7+ \sqrt{7}}$,$\sqrt{7+\sqrt{7+\sqrt{7}}}$,… converge?

The given sequence is $\sqrt{7}$,$\sqrt{7+ \sqrt{7}}$,$\sqrt{7+\sqrt{7+\sqrt{7}}}$,.....and so on. the sequence is increasing so to converge must be bounded above.Now looks like ...
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3answers
201 views

How could I find the sum of this infinite series by hand?

$$\sum_{n=1}^{\infty}\frac{(7n+32)3^n}{n(n+2)4^n}$$ Thank you!
4
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1answer
66 views

Elementary proof that $\sum_{k=1}^{\infty}{\frac{1}{m^{k^2}}}$ is irrational (for any integer $m > 1$)

I used similar technique as Fourier's proof of irrationality of $e$ https://en.wikipedia.org/wiki/Proof_that_e_is_irrational to show that this series is indeed an irrational number but I was wondering ...
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0answers
40 views

CEOI 1994 task Is there a sequence of n number where each sum of any p consecutive elements is positive and sum of any q is negative

Write a program which reads three positive integers $n, p, q.$ Decide whether or not there exists a sequence of n integers such that the sum of any p consecutive elements is positive and the sum of ...
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1answer
35 views

What does this point about triangular number mean

I was reading about triangular numbers from Wikipedia. I makes following point on the above web page: The number of line segments between closest pairs of dots in the triangle can be represented ...
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3answers
73 views

Is $\sum_{n=1}^\infty \frac{m}{(n+m)^2}$ bounded for all $m\in\mathbb{N}$?

I'm trying to figure out if there is a finite constant $C$ such that $\sum_{n=1}^\infty \frac{m}{(n+m)^2}\leq C$ for all $m\in\mathbb{N}$. I can see that $\sum_{n=1}^\infty ...
3
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5answers
131 views

How to find the MacLaurin series of $\frac{1}{1+e^x}$

Mathcad software gives me the answer as: $$ \frac{1}{1+e^x} = \frac{1}{2} -\frac{x}{4} +\frac{x^3}{48} -\frac{x^5}{480} +\cdots$$ I have no idea how it found that and i don't understand. What i did is ...
0
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1answer
16 views

Sequence m(k) with $\frac{m(k)}{3\cdot k\cdot log(k)}>0$ for $k\rightarrow\infty$

I'm looking for a sequence m(k) which fullfills the condition $\frac{m(k)}{3\cdot k\cdot log(k)}>0$ for $k\rightarrow\infty$. log(k) means the natural logarithm and m,k are positive integers. ...
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2answers
35 views

Engineering Mathematics Problem with Taylor's Series

This is a problem from Engineering Mathematics book by K.A. Stroud 7th edition, Exercise 18, Chapter 12 Further problems. It has been given in a physics manner, but it just requires manipulation of ...
2
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0answers
47 views

Positivity of an alternating series.

Greetings esteemed mathematicians. I've managed to prove that the following series \begin{equation} f_{\lambda}(\omega)= ...
2
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0answers
38 views

Expansion of Weierstrass elliptic function in second period

I would very much like to find expansions of the Weierstrass $\wp$ and $\zeta$-functions for small absolute values of the second period $\omega_2$. So, more precisely, I would like, for ...
0
votes
2answers
22 views

Accumulation/limit points

The problem is: For the sequences $(-1)^{n+1}\dfrac{n}{2n+5}$, $(-1)^{n+1}\dfrac{n}{2n^2+5}$ and $\dfrac{1}{\sqrt{n}}\cos\Bigl(1+\dfrac{1}{n}\Bigr)$ determine the accumulation ...
0
votes
1answer
24 views

Fourier Series Relation Time - Frequency

I want to study and understand the relation between time and frequency with the help of Fourier Series. Can you indicate me some papers, or some example?
1
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2answers
58 views

How to interpret $\sum_{n\in \mathbb N^{d}} \frac{1}{n^{p}};$ and when it is converges?

I know that: $\sum_{n\in \mathbb N} \frac{1 }{n^{p}}$ converges if $p>1$ and diverges if $p\leq 1$ My Question is: What is an analogue this in more than one variable (say $d$)? Does it make ...
1
vote
1answer
88 views

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was ...
0
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0answers
28 views

step by step solution of $y''+(x-1)y=0$ by Frobenius method

I've tried solving this equation, $y''+(x-1)y=0$, but honestly I'm not sure if I've done it right. Using Frobenius Method, I got $r_1=1$ and $r_2=0$ as the indicial roots which I think under case 3 as ...
5
votes
2answers
307 views

The longest sequence of numbers with a certain divisibility property

Definition - Denizen A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P} $; and it satisfies the following condition; ...