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

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0
votes
5answers
48 views

What is a quick way to establish that $\sum_{n=1}^\infty \frac {\log n}{n^{3/2}}$ converges?

What is a quick way to establish that $\sum_{n=1}^\infty \dfrac {\log n}{n^{3/2}}$ converges? Attempt: I proved this using the Integral Test but the integral test is usually a bit tedious. So, what ...
2
votes
1answer
20 views

Taylor series expansion of the function $f(x)=x \arctan x-0.5 \log(1+x^2)$ about the origin int the region {$|x|\le1$}

Find the Taylor series expansion of the function $\color {green}{f(x)=x \tan^{-1} x-0.5 \log(1+x^2)}$ about the origin int the region {$|x|\le1$} My effort: I know $\displaystyle \log ...
1
vote
1answer
28 views

Does $\Sigma_{n=1}^{\infty} \dfrac {\sqrt {2n-1}~ \log (4n+1)} {n(n+1)}$ converge?

Does $\Sigma_{n=1}^{\infty} \dfrac {\sqrt {2n-1}~ \log (4n+1)} {n(n+1)}$ converge? Attempt: I have been trying to use the comparison test for a while, but I can't find a suitable comparator. For ...
-3
votes
2answers
32 views

a Combinatorics problem in series [on hold]

Hey everyone i was having a problem with the following question: in how many ways is it possible to solve the following equation using natural numbers: $$ x_1+x_2+x_3...+x_{15}=300 $$ that for every ...
2
votes
5answers
125 views

$e^{x} > 1$ and $0 < e^{x} < 1$

So $$\exp(x) := \sum_{n=0}^{\infty} \frac {x^n} {n!}$$ How to prove that $\exp(x) > 1$ when $x > 0$ and moreover $\exp(x) < 1$ when $x<0$ Is it possible with induction? Or must I use ...
2
votes
4answers
71 views

$\sum_1^n 2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1} $ converge or not? [duplicate]

how to check if this converge? $$\sum_{n=1}^\infty a_n$$ $$a_n = 2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1}$$ what i did is to show that: $$a_n =2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1} > 2\sqrt{n} - ...
3
votes
1answer
171 views

How to compute this kind of limit? [duplicate]

Let $x_0=a,x_1=b$ $$x_{n+1}=\Big(1-\dfrac{1}{2n}\Big)x_n+\dfrac{ x_{n-1}}{2n}, n\ge1$$ Find $\lim x_n.$ If limit exist I can plug limit as $l$ to get a equation of $l$, whose root will be the ...
1
vote
0answers
27 views

Find function for data

I am working on an integer factorization problem, and I need to find a function for my data. I have tried several polynomial functions, but they did not work. I know that there are infinitely many ...
2
votes
2answers
55 views

$x_{n+2}=\frac 13 (x_n+2x_{n+1})$, then lim $x_n$=? [on hold]

Let $x_0=a,x_1=b$,if $x_{n+2}=\frac 13 (x_n+2x_{n+1})$, then $\lim x_n=?$ I see here a geometrical interpretation of the points of sequences. It is $ x_2$ the inner point 2:1 of a and b. ...
0
votes
0answers
33 views

Closed form of a series with sinh

Is there a simple form for following function (where $a$ and $b$ are constants)? Can it be simplified to a simple form if $a>>b$? $$ u(x) = \sum _{n=0}^{\infty } \frac{ \, (-1)^n ...
0
votes
2answers
26 views

How to solve this Geometric Series

I believe this is a geometric series and that is all the problem stated. If $0 < x < 1$, find $f(x) = 1 + x + x^2 + x^3 + x^4 + \cdots$
1
vote
0answers
20 views

What can be said about the limit of a converging infinite polynomial?

Suppose we have the following polynomial of infinite order: $f(x) = a_0+a_1x+a_2x^2+...=\sum_{n=0}^{\infty}a_nx^n$ Also suppose that $f(x)$ converges to some limit $f^*$ as $x\nearrow\infty$, i.e. ...
0
votes
1answer
36 views

How can I find the sum of an infinite series of products?

Background Today in my macroeconomics class my teacher taught us three concepts. The first is very simple: consumption $c$ is a linear function of national income $y$. Mathematically, $$c = My + ...
-4
votes
1answer
21 views

Expected value of a series [on hold]

Suppose you had a random series with a finite sum. What is the expected value of this sum?
1
vote
3answers
40 views

Generating series of integers with a specified sum

If I say that 6 positive integers were added together to get a total of 200. let count = 6 let sum = 200 I have 2 questions First of all, is there a formula for generating a list of all the possible ...
0
votes
0answers
38 views

$\lim \frac{x_1+ \dots +x_n}{y_n+\dots +y_n}=a$ [duplicate]

Suppose $y_n>0$ for all $n\in \Bbb{N}$, with $\sum y_n=+\infty$. If $\lim \frac{x_n}{y_n}=a$, prove that $\lim \frac{x_1+ \dots x_n}{y_n+\dots +y_n}=a$.
0
votes
1answer
16 views

Help Find A Sequence With The Property s(i) < s(i+1) XOR s(i+1) < s(i+2)

I'm looking for a sequence of positive integers with the property that, where $s(i)$ denotes the $i$th term, $s(i)<s(i+1)$ or $s(i+1)<s(i+2)$ but not both. The sequence also has the property ...
1
vote
3answers
30 views

$\sum_1^{\infty}\frac{\ln(n)}{n^{5/2}}$ and Cauchy condesation test

How can I prove that this series converges? I use cauchy condensation test but there is a problem because an is not non-increasing for $n=1,n=2$ .Should another test be used? ...
2
votes
2answers
38 views

Series convergence: $\sum_{n=1}^\infty\frac{\sin{\frac{n\pi}{2}}}{n^{2/3}}$

How can I test convergence for this series ? Comparison, limit test fail for me. $$\sum_{n=1}^\infty\frac{\sin{\frac{n\pi}{2}}}{n^{2/3}}$$
0
votes
0answers
17 views

Can someone give me an example of an M-Sequence please?

A sequence ${a_n}$ $(n\geq 0)$ of elements of $Z_p$ is said to be an m-sequence associated with $f$ if it satisfies the mod p linear recurrence relation; $$f_0{a_n} +f_1{a_{n+1}} ...
0
votes
3answers
47 views

how to find sum of the given series

How to find sum of the following series: $1+\dfrac{1}{3}\dfrac{1}{4}+\dfrac{1}{5}\dfrac{1}{4^2}+...$ The general term is $u_n=\frac{1}{2n+1}\frac{1}{4^n};n\geq 1$ Any help
-1
votes
1answer
50 views

Show that if $\lim_{n\rightarrow\infty}z_n=0$ then $\lim_{n\rightarrow\infty}(1+\frac{z_n}{n})^n$ [on hold]

Can you help me with the following problem. I dont have any idea how to start. Prove that if $\lim_{n\rightarrow\infty}z_n=0$ then $\lim_{n\rightarrow\infty}(1+\frac{z_n}{n})^n=1$.
1
vote
0answers
35 views

Test $\sum_{n=1}^{\infty}\Big(1-n \sin(\frac{1}{n})\Big)$ for convergence with asymptotic comparison

Suppose we want to test the following series for convergence $$\sum_{n=1}^{\infty}\Big(1-n \sin(\frac{1}{n})\Big)$$ I have found a solution that uses the asymptotic comparison test and uses the ...
-1
votes
1answer
39 views

Proving a sequence is Null, Help!

I have this question: Use the definition of a null sequence to prove that the sequence $\{a_n\}$ given by $a_n = \dfrac{2}{2n^2 -3}, n = 1, 2, \dots ,$ is null. So I know that we want to show for ...
2
votes
2answers
17 views

Finding the nth term in a recursive coupled equation.

I'm probably missing something simple, but if I have the recursive sequence: $$ a_{i+1} = \delta a_i+\lambda_1 b_i $$ $$ b_{i+1} = \lambda_2 a_i + \delta b_i $$ how would I find a formula for $a_n$, ...
2
votes
1answer
75 views

Is this series: $\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}}$ convergent?

How can I show that the following series is convergent or divergent ? $$\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}},x\in \mathbb{R}$$ I want to use Abel-Dirichlet criteria. I've ...
1
vote
4answers
50 views

How to check the convergence of $\sum_{n=1}^\infty\frac{\ln(n^2+n+1)}{n^{3/2}}$

There is an example of the Limit Comparison test on my textbook, and it finds the convergence of this series: $$\sum_{n=1}^\infty\frac{\ln(n^2+n+1)}{n^{3/2}}$$ It starts off with the limit ...
1
vote
1answer
34 views

How to find $ \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^2}$?

Let $f$ be a $2\pi$-periodic function whose restriction on $[-\pi, \pi]$ is $f(x)_{[-\pi, \pi]} = |x|$ It is easy to see that its fourier series converges uniformily to $f$ and is $$f(x) = \frac \pi2 ...
0
votes
2answers
45 views

Is $\sin\left[{\pi \cdot \frac {1}{\sqrt{n^2+1}+n}}\right]$ decreasing?

How can I show that $$\sin\left({\pi \cdot \frac {1}{\sqrt{n^2+1}+n}}\right)$$ is decreasing for $n>1$? I think I have to show that the expression from inside the $\sin$ expression is between $\pi ...
2
votes
1answer
37 views

Computing lim sup and lim inf of $\exp(n\sin(\frac{n\pi}{2}))+\exp(\frac{1}{n}\cos(\frac{n\pi}{2}))$ and $\cosh(n\sin(\frac{n²+1}{n}\frac{\pi}{2}))$?

It's the first time I encounter lim sup and lim inf and I only just know about their definitions. I have difficulties finding out about lim sup and lim inf of the following sequences ...
0
votes
1answer
31 views

Regarding sup and inf of a continuous function

Suppose $f:\mathbb R\to \mathbb R$ be a continuous function such that $\lim\limits_{x\to \infty}f(x)=0=\lim\limits_{x\to -\infty} f(x)$. Then I want to show that $f$ is bounded and attains at least ...
0
votes
1answer
36 views

Symmetric difference and convergence of sequence of sets

I have two question regards to symmetric different and a convergent sequence of a set:- if we have a sequence of sets $\{X_i\}$.then how to show that:- $\{X_i\}$ is convergent if and only if if for ...
0
votes
1answer
16 views

Alternative representation of time series

In a paper I am reading, it refers to the following time series model: $$ Y_t=\rho Y_{t-1}+e_t $$ Where $ \lvert\rho\rvert < 1$ It goes on to say that this process can be represented in the ...
0
votes
1answer
47 views

$\sum_{i=1}^{\infty} n^2 (n+1)!$ [closed]

How to find the sum of $\sum_{i=1}^{\infty} n^2 (n+1)!$
2
votes
3answers
53 views

Prove that {$r_n$} satisfies $r_n = 7r_{n-1} - 10r_{n-2}, n \geq 2$

Problem: For the sequence $r$ defined by $$r_n = 3 \cdot 2^n - 4 \cdot 5^n, \ \ \ n \geq 0$$ Prove that {$r_n$} satisfies $$r_n = 7r_{n-1} - 10r_{n-2}, \ \ \ n \geq 2$$ Can this problem be ...
1
vote
1answer
31 views

Differentiating under the summation

I saw on the Wikipedia page for differentiation under the integral that it could also be applied to summations. Here is the link: ...
1
vote
0answers
58 views

What is an elegant way to express $(-1)^k$

In computation of series, a lot of times you will find a term $(-1)^k$ jutting out in an otherwise easy to remember expression. Is there some interesting way to write $(-1)^k$ that may help in ...
2
votes
4answers
117 views

The convergence of $\sum_{n=2}^{\infty} {1\over \ln(n!)}$

Check the convergence of $\sum_{n=2}^{\infty} {1\over \ln(n!)}$. I tried D'Alembert's test... Cauchy's test seems too intricate... I can't seem to understand what I should do here...
0
votes
0answers
32 views

Solving a recurrence relation using the substitution method

Consider the recursive function $f(n)=3f(n/4)+2n $, $f(16)=32.$ Where n is always a power of 4 greater than 16. We must find a closed form utilizing substitutions. So, after one substitution, f(n) ...
1
vote
2answers
75 views

Determine if the following series is convergent or divergent $\sum_{n=1}^{\infty}\frac{ \sin{nx}}{n}$

I've been doing exercises with series for some time and now I've got an exercise that I'm supposed to solve using Abel or Dirichlet criteria. The problem is : Determine if the following series ...
1
vote
2answers
27 views

Converges or diverges: $\sum_{n=1}^{\infty}\left [\arctan{\frac{(-1)^n}{n}}+{(-1)^n\over n^2}\right]$

How can I show that the following series converges or diverges ? $$\sum_{n=1}^{\infty}\left [\arctan{\frac{(-1)^n}{n}}+{(-1)^n\over n^2}\right]$$ $\sum_{n=1}^{\infty}\left ...
0
votes
2answers
42 views

Generalized geometric series value

Why the value of the following summation: $$1 + \sum_{k=1}^{n}\bigg(1- \frac{76}{i}\bigg)^k= \frac{i}{76}$$ is $\frac{i}{76}$? $\quad i$ is a positive constant.
1
vote
2answers
34 views

Determine if it converges or diverges : $\sum_{n=1}^{\infty} \frac {2^n \cdot n!}{1\cdot2\cdots (2n-1)}\cdot \frac{1}{\sqrt{2n+1}}$

Here's the series: $$\sum_{n=1}^\infty \frac {2^n \cdot n!}{1\cdot2\cdots (2n-1)}\cdot \frac{1}{\sqrt{2n+1}}$$ Does it converge or diverge ? Thanks
3
votes
1answer
25 views

$l_2$ sequence, series with square root

I'm trying to prove that the following functional is continuous: $$\phi : \mathcal{l}_2 \ni \{x_n \} \rightarrow \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}x_{3n} - \sum_{n=1}^{\infty} \frac{1}{n}x_{2n} ...
0
votes
1answer
51 views

Convergence of: $\sum_{n=1}^{\infty} \frac{2+(-1)^n}{2^n+(-1)^n} $

I've tried solving this exercise but got stuck on a big expression that I could not untangle. I've obtainded the following thing: $$\lim_{n \to \infty} n\frac{2 \cdot 2^n +3 \cdot 2^n\cdot ...
3
votes
2answers
61 views

Is $\lim\sup=\sup\lim$?

Assume $(a_n(x))_{n=1}^{\infty}$ is a bounded sequence in $\mathbb R$, when $x$ is $\in\mathbb R$ and is relevant to the sequence in some way that doesn't really interest us in my question. Assume ...
1
vote
2answers
32 views

Indeterminate form as a series

We know that $0 \times \infty$ is an indeterminate form. However, is it equivalent to $0 + 0 + 0 + \cdots$? If yes, why we do not consider $\displaystyle \sum_{n = 0}^\infty 0$ an indeterminate form? ...
1
vote
1answer
15 views

Alternating series, even terms, factorial, boundedness

I need to determine whether there exists $M>0$ such that $$| \sum_{n=0}^{\infty} \frac{(-1)^n x_{2n}}{\sqrt{n!}}| \le M \sqrt{\sum_{n=0}^{\infty}|x_n|^2}$$ $\{x_n\} \in \mathcal{l}^2, \ \ x_n \in ...
-1
votes
1answer
56 views

Maximising $\log_{2015} ( a_{2015} ) - \log_{2014} ( a_{2014} )$ given the properties of the sequence [closed]

Let $a_1, a_2, a_3 , \ldots$ be a sequence of positive real numbers such that For all positive integers $m$ and $n$ we have $a_{mn} = a_ma_n$, and There exists a positive real number ...
0
votes
1answer
21 views

Construct a non-constant sequence

Let $$S^{n}_{r}=\bigl\{{\overline{x}\in\mathbb{R}^{n+1}\;:\; \|{\overline{x}}\|=r}\bigr\}$$ thn $n-$ sphere with radius $r$ where $\|{\cdot}\|$ is the usual norm in $\mathbb{R}^{n+1}$ and a point ...