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

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-3
votes
0answers
27 views

help with showing a series is divergent [closed]

I tried unsuccessfully to show by convergence tests that the series $$\sum_{n=1}^\infty{\ln^nn\over n^2}$$ is divergent , cant seem to find a way. help would be very appreciated , thanks in ...
2
votes
2answers
20 views

Proving two Recurrence Relations

I encountered this problem in the International baccalaureate Higher Level math book, and my teacher could not help. If anyone could please take the time to look at it, that would be great. I need ...
-1
votes
1answer
60 views

How can I find out if a non-convergent series is “indeterminate” (that is, “oscillating”) or “divergent”?

Definitions: Given a sequence $\{a_n\}$, define $$s_n= \sum_{j=0}^n a_j.$$ The sequence $\{s_n\}$ is called the series of partial sums of $\{a_n\}$. A series is convergent if $\{s_n\}$ has ...
0
votes
1answer
28 views

Convergence of $\sum_{n=0}^\infty \frac{\sqrt{1+x^n}}{x^n}$ in the case $x<0$ and an analogous problem with $\sum_{n=0}^\infty \frac{x^n}{2+x^n}$

Let $$\sum_{n=0}^\infty \frac{\sqrt{1+x^n}}{x^n}.$$ My first question is: for what values of $x$ is this series possible? I can only say that it is not defined for $x = 0$, but are there other ...
-6
votes
0answers
32 views

Help and solve for me on Sequences [closed]

solve for me this sequence. The number of the sequence is 17, if the second number is 3, what is the first number in that sequence
0
votes
0answers
20 views

Algorithm for filling in points around a circle with increasing density

The aim of this question is to decide on the order in which to download a series of high-resolution files that together represent a 720° rotation around an animating object. When all the files are ...
-2
votes
0answers
32 views

$\textrm{lim sup}$ of a function [closed]

I need to compute a limit: Given a series with the sum equal to $1$, I need to compute $\textrm{lim sup}\ (A_n)^{1/n}$. Since the sum is $1$, I assume that the value $A_n$ which actually should tend ...
0
votes
0answers
25 views

Algorithm for generating a series of contrasting colours

On a computer screen, colours can be defined as having 0-255 units of red, green and blue. This creates a 3-dimensional colour space with $256^3$ different colours, from 0-0-0 for black to 255-255-255 ...
1
vote
1answer
79 views

For which values of $q$ and $\alpha$ does $\sum_{n=0}^\infty q^{n^\alpha}$ converge?

Let us consider the series $$\sum_{n=0}^\infty q^{n^\alpha}, \qquad \qquad\qquad (*)$$ with $\alpha,\in \mathbb{Q}, n \in \mathbb{N}$. I know that the geometric series converges if $|q|<1$, but ...
2
votes
3answers
77 views

Does $\sum_{n=2}^\infty\frac{1}{n\ln n!}$ converge?

Let $$\sum_{n=2}^\infty \frac{1}{n\ln n!}.$$ It is equal to $$\sum_{n=2}^\infty \frac{1}{n(\ln n + \ln(n-1) +...+ ln(2))}.$$ But now what should I do to prove that it converges? (I have tried root ...
0
votes
1answer
25 views

How to show this is decreasing

I'd like to show $$\sum_{i=1}^n \frac{1}{i((n+1)-i)} $$ is decreasing for n>1, which is Cauchy product of $$\sum_{i=1}^n \frac{1}{i}$$ Numerical computation until n=50 shows it's decreasing but I ...
0
votes
1answer
34 views

Calculate the sum of the series $\sum_{n=0}^{\infty} \frac{1}{a_{n}\cdot a_{n+1}\cdot\ldots\cdot a_{n+7}}$, where $a_k = ak + b$

Let $a_k = ak + b$; define the following series: $$\sum_{n=0}^{\infty} \frac{1}{a_{n}\cdot a_{n+1}\cdot\ldots\cdot a_{n+7}}.$$ I have to prove that this series converges and I have to find its sum. ...
1
vote
2answers
28 views

Series definied by recurrence relation

Define a sequence recursively by $$\large{a_1 = \sqrt{2},a_{n+1} = \sqrt{2 + a_n}}$$ (a) By induction or otherwise, show that the sequence is increasing and bounded above by 3. Apply the monotonic ...
0
votes
2answers
33 views

Study: $\sum_{n=1}^\infty (\sin(\sin n))^n$, $\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$, and $\sum_{n=1}^\infty \frac{ \sin (x^n)}{(1+x)^n} $

Let $x \in \mathbb{R}$. I have to study the convergence of the following three series: $$\sum_{n=1}^\infty (\sin(\sin n))^n$$ $$\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$$ ...
5
votes
1answer
49 views

Why is the Cauchy product of two convergent (but not absolutely) series either convergent or indeterminate (but does not converge to infinity)?

It is well-known that the Cauchy product of two absolutely convergent series is absolutely convergent. However, my professor added (without giving a proof) that if the series are convergent ...
0
votes
1answer
18 views

Sequence of holomorphic functions and approximation by polynomials.

Let $\Omega=\{ z\in \mathbb{C}:$ $Im$ $z>0,$ $|z|>1\}\cup\{z \in \mathbb{C}:$ $Im$ $z<0$ $|z|>1\}$ I know that since $\hat{\mathbb{C}}\setminus \Omega$ is connected there's a sequence of ...
2
votes
0answers
15 views

Upper bounding Lerch zeta function

Let $\Phi\left(z,s,a\right) $ be a Lerch Trascendent. $$\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}}{1-ze^{-t}}dt.$$ Can we upper bound the above in ...
1
vote
0answers
23 views

Cauchy Criteria for Series

We know that the Cauchy Criterion of a series is as follow: Theorem: A series $\sum\limits_{i=1}^{\infty}x_i$ converges iff for all $\epsilon>0$ there is an $N\in \mathbb{N}$ so that for all ...
2
votes
3answers
35 views

Number sequence - arithmetic sequence difference constant - find formula

I searched a way to find out a formula to predict the nth number of a given sequence, but I did not find a way matching my case. Arithmetic sequence: I read that a good way is to find the constant ...
6
votes
1answer
69 views

A limit with the harmonic series

How can we prove the following (similar) limits? $$\sum_{k=1}^n \frac{1}{k} (\ln 2 - \frac{1}{n+2} - \frac{1}{n+3} - \cdots -\frac{1}{2n + 2}) \to 0. $$ $$\sum_{k=1}^n \frac{1}{k} (\ln 3 - ...
3
votes
0answers
41 views

Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
4
votes
1answer
38 views

Order of Growth of a Sum

Let $n>k$ and $r$ be arbitrary positive integers. Define $q=k/n$. I want to show that $$ \sum_{i=0}^{rk} \binom{rn}{i}q^i(1-q)^{rn-i}(rk-i)=\Theta(\sqrt{r}) $$ as $r\rightarrow \infty$. I've ...
0
votes
2answers
43 views

What is the general formula $t_n$ for this sequence? [closed]

$1, 4, 8, 13, 19, \dots$ What is the general formula or term for this sequence? How do you find it without a common ratio or difference? Please explain.
1
vote
1answer
47 views

Proving $(n+1)c^{1/(n+1)} - nc^{1/n} \le 1$ from first principles

Is it possible to prove that \begin{align*} (n+1)c^{1/(n+1)} - nc^{1/n} \le 1 \qquad c \in \mathbb{R}_+, n \in \mathbb{N} \end{align*} using only elementary techniques? (No calculus, no appeasement to ...
1
vote
1answer
28 views

Why is the identity function from $\Bbb R$ with the Euclidean metric to $\Bbb R$ with the discrete metric not continuous?

Using only the definition of sequential continuity, show an example that $f(x) = x: \Bbb R \to \Bbb R'$ is not continuous, where $\Bbb R'$ has the discrete topology. So the definition of ...
0
votes
1answer
26 views

Is it possible to exhibit a collection of sets

Let a subset $D$ of the natural numbers be called convergent or divergent when the associated series $\sum_{d \in D} \frac{1}{d}$ converges or diverges. Define a topology on $\Bbb{N}$ by defining the ...
2
votes
3answers
54 views

Convergence of $\sum_{n=0}^\infty \frac{\ln(1+2^n)}{n^2+x^{2n}}$

Let $x \in \mathbb{R}$. Define the series: $$\sum_{n=0}^\infty \frac{\ln(1+2^n)}{n^2+x^{2n}}.$$ For what $x$ does it converge? It clearly has positive terms. The ratio and root tests seem ...
22
votes
5answers
2k views

Is there any explanation for the repetitions after decimal point on divisions like 24/7

I was trying to divide 24 by 7 using a pen and a paper. After I had no more space on my checkerboard paper, I decided to put it on a calculator. The calculator returned 3.428571428571429 and I ...
2
votes
0answers
43 views

For which values of $\alpha, \beta, x \in \mathbb{R}, x \geq 0$, does the series $\sum_{n=1}^\infty n^ \alpha x^{n^{\beta}}$ converge?

I have to study, for $\alpha, \beta, x \in \mathbb{R}$, $x \geq 0$, the convergence/divergence/irregularity (i.e., when the limit of the $N$-th partial sum does not exist) of the following series: ...
1
vote
2answers
71 views

For which $x \in \mathbb{R}$ does the series $\sum_{n=1}^\infty \frac{x^n}{n!}$ converge?

One problem of my exercise book asks for which $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \frac{x^n}{n!}.$$ The answer given by the exercise book is $|x|\leq 1$, but ...
6
votes
1answer
52 views

For which $x\in \mathbb{R}$ does $\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$ converge?

I have to study for which values of $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$$ I was only able to say that the ...
5
votes
1answer
77 views

For what values of $x$ does the series $\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$ converge?

I have to study the values of $x$ for which $$\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$$ converges. First we say that we must have $x>0$. Then, I have started by rewriting the series as ...
6
votes
4answers
49 views

For which values of $\alpha \in \mathbb{R}$, does the series $\sum_{n=1}^\infty n^\alpha(\sqrt{n+1} - 2 \sqrt{n} + \sqrt{n-1})$ converge?

How do I study for which values of $\alpha \in \mathbb{R}$ the following series converges? (I have some troubles because of the form [$\infty - \infty$] that arises when taking the limit.) ...
0
votes
1answer
32 views

Hilbert's inequality for $\left|\sum_{n,m}a_n \bar a_m\right|$.

We know that, an Hilbert's inequality states $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ Give $a_n, b_n$ two sequences of complex numbers. Then write an inequality ...
1
vote
2answers
42 views

Limited partial sum of $\displaystyle \sum _{n=1} ^{k} \cos(nx)$ are limited?

I'm wondering if it's true that $\displaystyle \sum _{n=1} ^{k} \cos(nx)$ has limited partial sum. I know it has representation $\displaystyle ...
2
votes
1answer
39 views

How to tell if a series diverges or is indeterminate? Study of some cases of $\sum_{n=1}^\infty3^n (1+\frac{1}{n})^{n^2}k^n$

Suppose we have a series dependent on a parameter. For example: $$\sum_{n=1}^\infty3^n (1+\frac{1}{n})^{n^2}k^n.$$ By root test, we know that this series absolutely converges (hence converges) if ...
-4
votes
1answer
56 views

Is this series computable? [duplicate]

I would like to compute the value of this series: \begin{equation*} \sum_{n = 0}^{+ \infty} n . e^{- \alpha n} \end{equation*} Where $\alpha$ is a constant.
1
vote
0answers
34 views

Arithmetic progression Find first term and common difference when sum of 10 terms and the 8th term is given

Sigma is a car company that sell cars. Sigma sells $x$ cars in the first month and its sales increase constantly by $y$ cars every subsequent month. It sells $96$ cars in the $8^{th}$ month and the ...
0
votes
2answers
60 views

prove or disprove convergence [duplicate]

Im trying to prove or disprove the following , but I am having a hard time. It seems that the statement is true , but I have no idea how to prove it. If \begin{equation*} \sum_{n} a_{n}^2 ...
2
votes
2answers
54 views

Sum of the series $\tan^{-1}\frac{4}{4n^2+3}$

Find the value of $$\sum^{n=k}_{n=1}\tan^{-1}\frac{4}{4n^2+3}$$ I tried multiplying numerator and denominator by $n^2$, but got nothing. How do I split the term inside $\tan^{-1}$?
0
votes
2answers
74 views

Limit of $\frac{\binom{2n}{n+2}n^{n-2}(n.(n+1) \cdots 2n)}{n^{2n}}$ as $n\to \infty$.

Limit of $$\frac{\binom{2n}{n+2}n^{n-2}(n.(n+1) \cdots 2n)}{n^{2n}}$$ as $n\to \infty$ Does it go to zero? WolframAlpha says it does, but it does not make sense since \begin{equation*} ...
0
votes
1answer
41 views

Proof for Mean of Geometric Distribution

I am studying the proof for the mean of the Geometric Distribution http://www.math.uah.edu/stat/bernoulli/Geometric.html (The first arrow on Point No. 8 on the first page). It seems to be an arithco ...
0
votes
0answers
17 views

Probability of loosing a consecutive sequence

Use Case A device needs need to receive keep alive messages which keeps the device connection open for 30 seconds. The line should be able to cope with $5$% continuous packet loss. Configuration ...
0
votes
2answers
44 views

Summation of a series

I am interested to get a tight upper bound on the summation of the following series $S$. $\displaystyle S=\sum_{i=1}^N\frac{e^{-\alpha i}}{(k+i)^d}$ for integers $k,N \geq 1$, and positive reals $d$ ...
1
vote
0answers
22 views

Convergent sequences doubt from Rudin

What does the following mean: The sequence $<1/n>$ converges in $\mathbb R$ (to 0) but fails to converge in the set of all positive real numbers [with $d(x,y) = |x-y|$]. Reading Rudin's ...
1
vote
2answers
39 views

Is there any way to find sum of sequence generated by formula?

I have such sequence: $2^2$, $7^2$, $12^2$, $17^2$, $22^2$, ... I found a formula to generate n-th term: $(5\cdot n+2)^2$ And I need to find a sum of those numbers: 4, 49, 144, 289, 484, ... Can ...
-4
votes
1answer
33 views

real analysis diiferentiation [duplicate]

I have no idea how to approach this question. Please help. How do I use definition of improper integral to solve it?
0
votes
1answer
24 views

find interval of convergence for power series of the following function

I have found the power series for the following function, but I cannot figure out how to find it's interval of convergence due to the $x^2$ remaining. The function is this (I'm so sorry I have not ...
1
vote
0answers
33 views

Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the ...
0
votes
0answers
33 views

series and dyadics [duplicate]

I'm in Calc II and am unfamiliar with what dyadics is but my teacher said that it's possible to find the sum of $\frac{\cos n}{n^2}$ by using dyadics. Would you mind laying out a step by step? ...