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

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0
votes
1answer
25 views

Find a Taylor series around $x=0$

I don't know how to find the Taylor series around $x=0$ for: $$f(x)=\frac{\tan(2x)-\arctan(4\sinh(x))}{\sin(x^{2})}$$ Thank you in advance.
0
votes
2answers
82 views

convergence of $(1+\frac{1}{2})(1+\frac{1}{4})\ldots(1+\frac{1}{2^n})$

I was given a problem of testing sequence convergence. The sequence is defined as: $$x_n= (1+\frac{1}{2})(1+\frac{1}{4})\ldots(1+\frac{1}{2^n})$$ My first idea was to define $y_n$ as follows: $$y_n = ...
2
votes
3answers
77 views

Simplify the sum $ \sum_{k=1}^{\infty} (\frac{1}{2})^kk $

I need some help simplifying this sum: $$ \sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^kk $$ I have a feeling it's some basic series thing that I'm forgetting, but I need help nonetheless.
5
votes
2answers
100 views

What is the general term of $a_{n+1}=\frac{2a_n-1}{5a_n-1} \ , \ \ a_1=1$?

I've struggled to solve this exercise $$a_{n+1}=\frac{2a_n-1}{5a_n-1}\ , \ \ a_1=1$$ $$b_{n+1}=(5a_n-1)b_n \ , \ \ b_1=1$$ Find $b_{\ 40}$ . $$$$ I thought 'taking inverse' will be ...
1
vote
1answer
45 views

What kind of sequence is that ($1+2+2^2+\cdots+2^k$) and how it can be expressed in a short way?

I am curious what kind of sequence is that $$1+2+2^2+2^3 +\cdots+2^{k-1}$$ and how it can be simplified or expressed in some short way... In the classroom we expressed it as $2^{k-1}$ over something ...
2
votes
0answers
41 views

How can I calculate the sum of the following series? [duplicate]

I am trying to calculate the sum of S=(1-1/2 +1/3-1/4+...). I used wolfram Alpha but the answer makes no sense to me. Thank you in advance.
0
votes
0answers
14 views

Limit of sequence (limit of Bilateral sequence)

I have a question related sequence and limits of sequence. From definition we know that sequence is a function whose domain is natural number.Then we called a sequence (a_n) converges if for every ...
1
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0answers
23 views

Convergence of infinite products

I wonder, parallel to the theory of summability of infinite series is there a theory for infinite products? Is there any generalized convergence method (such as Cesaro and Abel summability) for the ...
2
votes
2answers
26 views

Measure of a modified Cantor set

Suppose a modified Cantor set: Starting with $E_0 = [0,1]$, we delete the middle interval of length $1/3$, then we delete the middle intervals of length $1/15$, and so on; in each step we delete from ...
1
vote
1answer
36 views

Find polynomial f(n) such that for all integers $n$ $\geq 1$, we have

Find polynomial f(n) such that for all integers $n \geq 1$, we have $3\left( 1\cdot2 + 2\cdot3 + \ldots + n(n+1) \right) = f(n)$. Write f(n) as a polynomial with terms in descending order of $n$.
4
votes
1answer
29 views

If $a_n$ and $b_n$ are equivalent sequences and $a_n$ is bounded then so is $b_n$.

This is what i know; If $(a_n)$ is an infinite sequence of which is bounded then we can say; $|a_i| < M $ for all $i \geq 0.$ since $a_n$ and $b_n$ are equivalent sequences, we can say that for ...
0
votes
2answers
25 views

Series with floor function - convergent?

I am trying to figure out where does the following series converge to as $n$ goes to infinity (if it doest at all) $$\frac{1}{n} \sum^{n}_{t=\lfloor \rho n \rfloor +1} \frac{n}{t}$$ where $\rho$ is ...
1
vote
2answers
58 views

Question about series convergence $\sum_{n=1}^\infty \frac{1}{n}$ and $\sum_{n=1}^\infty \frac{1}{n^2}$

So I have been playing around with convergent series recently and I still have a hard time understanding why $\sum_{n=1}^\infty \frac{1}{n}$ diverges and $\sum_{n=1}^\infty \frac{1}{n^2}$ converges. ...
0
votes
1answer
42 views

Calculate this infinite sum [duplicate]

$$s= \sum_{n=1}^\infty \frac{n+3}{(2^n)(n+1)(n+2)}$$ Any method to calculate this type of infinite sums?
0
votes
0answers
35 views

Is this approach to the Collatz conjecture flawed?

This question has 3 parts. 1/ outlines a "shadow function"; 2/ uses this shadow function to show there are no non-trivial loops; 3/ uses the notion of "combs" to show there are no sequences ...
1
vote
0answers
35 views

Closed form for the summation $\sum_{k=1}^n\dfrac{1}{r^{k^2}}$

Is there any closed form for the finite sum $$\sum_{k=1}^n\dfrac{1}{r^{k^2}}$$ or infinite sum ( when $|r|<1$) $$\sum_{k=1}^\infty\dfrac{1}{r^{k^2}} ?$$ While solving this problem, I found this ...
0
votes
3answers
37 views

Convergence of sequence s.t $|a_{n+1}-a_n|\le \frac{n^2}{2^n}$

Let $a_n $ be sequence of real numbers such that $|a_{n+1}-a_n|\le \frac{n^2}{2^n} $for all n $\in $ N. Then $ a_n $ is convergent $ a_n $ is bounded but not convergent $ a_n $ has 2 limit points. ...
0
votes
1answer
25 views

How to solve the recursion $f(n+2)=3f(n+1)-2f(n)+5$?

$$f(n+2)=3f(n+1)-2f(n)+5, \text{ with } f(1)=4, f(2)=5\\ f(n+2)=3f(n+1)-2f(n)+n, \text{ with } f(1)=4, f(2)=5$$ I can't find anywhere the solution for sequences of this type and am unable to figure ...
2
votes
1answer
54 views

closed form for a double sum

How can I prove that $$\underset{k\geq1}{\sum}\left(\underset{m=-\infty}{\overset{\infty}{\sum}}\frac{\left(-1\right)^{m}}{\left(2k-1\right)^{2}+m^{2}}\right)=\frac{\pi\log\left(2\right)}{8}\,?$$I ...
0
votes
1answer
30 views

If there is a positive constant $r$ such that $c_n \geq r >0 ~\forall~n\in \mathbb N$, then prove that $\sum_{n=1}^\infty a_n $ converges

Let $\{a_n\}, \{b_n\}$ be positive sequences. Let $c_n= b_n-\dfrac {b_{n+1}a_{n+1}} {a_n}$. Prove that : If there is a positive constant $r$ such that $c_n \geq r >0 ~\forall~n\in \mathbb N$, ...
-2
votes
1answer
20 views

Uniformly convergent & point convergent

Let ${({f_n})_n}$ is a sequence that ${f_n}(x)=tan^{-1}(nx), x\in [0,\infty)$. Prove for every $[a,b]$ that $a>0$ is Uniformly convergent and on $[0,b]$ just point wise convergent.
2
votes
2answers
31 views

Convergence of $\sum_{n=1}^\infty \dfrac {n^3[\sqrt 2 + (-1)^n]^n}{1.05^n} $

Test convergence of $\sum_{n=1}^\infty \dfrac {n^3[\sqrt 2 + (-1)^n]^n}{1.05^n} $ Attempt: By using the the $n^{th}$ root test : $\lim_{n \rightarrow \infty} a_n^{1/n} = \lim_{n \rightarrow \infty} ...
-1
votes
1answer
42 views

absolutely convergent & conditionally convergent [on hold]

Prove that $$\sum_{n=1}^{\infty}\left(\sum_{m=1}^{n}\frac{{{1}}}{{{m}}}\right)\frac{{{\sin(nx)}}}{{{n}}} $$ for $x = {k\pi}$ , $k\in \mathbb{Z} $ is absolutely convergent. & for $x ...
-4
votes
0answers
77 views

Explain about absolute convergence and convergence [on hold]

$$\sum_{n=1}^{\infty}\frac{{{(-1)^{n}\sin (n)}}}{{{n}}} $$ Explain about absolutely convergent and convergent. Thx a lot.
1
vote
0answers
18 views

how to prove $\Phi(t)$ is divergent when $Im(t)=\pi/2$?

The Riemann $\Xi(z)$ function admits a Fourier transform of a even kernel $\Phi(t)=3e^{5t/4}\theta'(e^{t})+2e^{9t/4}\theta''(e^{t})$. Here $\theta(z)$ is the Jacobi theta function. ...
0
votes
2answers
48 views

Prove$\sum_{n=0}^{\infty} \frac{n}{(n+1)^2}-\frac{1}{n+2} $ is convergent [on hold]

prove $$\sum_{n=0}^{\infty}\frac{{{n}}}{{{(n+1)^{2}}}}-\frac{{{1}}}{{{(n+2)}}} $$ is convergent.
0
votes
2answers
41 views

$ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent. [on hold]

prove that if $ \sum_{n=1}^{\infty} {a_n} $ is absolutely convergent $ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent.
0
votes
0answers
22 views

Prove the convergence and find limit [duplicate]

Let $x_n$ be a sequence with general formula: $$x_{n+1}=\sqrt{2+x_n}$$ I am supposed to prove it's convergence and find the limit as $n\to +\infty$ I thought of proving this sequence to be monotone ...
0
votes
1answer
22 views

prove that Radius of convergence is 1 [on hold]

Let's assume that $${\left\{ {{a_n}} \right\}_{n \in {\Bbb N}}}$$ is a positive sequence number. and let $$ \mathop {\lim }\limits_{k \to \infty }{A_k}=\sum_{n=0}^{k} {a_n} = \infty $$ if $$ \mathop ...
0
votes
1answer
21 views

Divisibility of binomial coefficients

I have got this series of binomial coefficients - $${2n\choose 0}+3\times{2n\choose 2}+3^{2}\times{2n\choose 4}+\ldots +3^{n}\times{2n\choose 2n}$$ I have to prove this to be divisble by ...
1
vote
1answer
27 views

Showing that the complete elliptic integral of the second kind can be represented by a particular series

I'm trying to show that the complete elliptic integral of the second kind can be represented as: $$E(k)=\frac{\pi}2 ...
0
votes
1answer
29 views

Convergence of $\sum_{n=1}^\infty{\frac{\ln({3n^2 +4n+5})}{n^{4/3}}}$

How can I test convergence for this series? I used limit comparison test with $\frac{1}{n^{5/4}}$ and seems to work, however I am looking for a simpler solution. $$\sum_{n=1}^\infty{\frac{\ln({3n^2 ...
4
votes
2answers
93 views

A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1(x)_n dx$

I would like a numerical evaluation of the series $$S=\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x(x+1)\cdots(x+n-1) dx$$ to five significant digits. I've used Mathematica, but some ...
1
vote
2answers
47 views

Find the sum of the series $\sum \limits_{n=1}^{\infty} \frac{2^{-n}(n+3)}{(n+1)(n+2)}$ [on hold]

Find the sum of $\sum \limits_{n=1}^{\infty} \frac{2^{-n}(n+3)}{(n+1)(n+2)}$
1
vote
1answer
58 views

Is this equality true? Why? Why not?

Let $$ \lim_{a\to 0} \frac{1}{2} \left( \left( \sum_{n=-\infty}^\infty \frac{1}{(n+a)^2} - \frac{1}{a^2} \right) \right) = \sum_{n=1}^\infty \frac{1}{n^2}$$ I already know that ...
-2
votes
1answer
23 views

Recurrence relations and initial conditions [on hold]

I couldn't figure out how to do the super/subscript, hence the photo.
-1
votes
1answer
34 views

How to find the formula of S3n? [on hold]

I have series $$1+2+3+4+4+3+5+5+6+...$$ if $n \in N$ then the formula of $S_{3n}=...$ Even I'm confused the pattern of the sequence. Anyone can help me? I was just given one clue, this is arithmetic ...
4
votes
2answers
50 views

Prove that the sequence $p_i$ is bounded

Let $p_1,p_2,...$ be a sequence of natural numbers. $p_1$ and $p_2$ are prime and $p_n$ for $n\ge 3$ is the largest prime divisor of $p_{n-1}+p_{n-2}+2014$. Prove that $(p_n)$ is bounded.
1
vote
3answers
46 views

Problem with sequences and series problems

Having studied Mathematics as my principle subject in my graduation, I do remember in the topics of sequences and series, then practically any number can be present in sequence, provided there is an ...
2
votes
1answer
17 views

Prove the series converges uniformly at $[x_0, \infty)$

Let $\sum_{n=0}^\infty a_ne^{-\lambda_n x}$, where $0 < \lambda_n < \lambda_{n+1}$. It is given that the series converges at $x_0$. Prove that the series converges uniformly at $[x_0,\infty)$. ...
1
vote
1answer
70 views

Find all natural sequences $a_n=a_{a_{n-1}}+a_{a_{n+1}}$

Find all natural sequences for which holds $a_n=a_{a_{n-1}}+a_{a_{a+1}}$ a) for all natural numbers $n\ge 2$ b) for all natural numbers $n\ge 3$ I tried to do something with the caracteristic ...
1
vote
1answer
89 views

Prove that this sequence of integers is on average equal to zero.

Consider the sequence $\{a(n)\}_{n\in\mathbb{N}^*}$ that is defined by the Dirichlet series: $$\zeta (s)^2\cdot\left(1-\frac{1}{2^{s-1}}-\frac{1}{3^{s-1}}+\frac{1}{6^{s-1}}\right)=\sum_{n\geq ...
2
votes
1answer
22 views

Prove that space of Laurent series is not Hilbert

Let $z_0 \in \mathbb{C}, s>0,T(z_0,s):=\{z\in\mathbb{C}:|z-z_0|=s\}$ and let $V=V(z_0,s)$ be a vector space over field $\mathbb{C}$ of all Laurent series that are uniformly and absolutely ...
0
votes
3answers
56 views

Series proof $\sum_1^\infty|a_n|<\infty$ then show that $\sum_1^\infty{a_n^2}<\infty$

Iam stuck with this proof. There seems to be no property to help. If $\sum_1^\infty|a_n|<\infty$ then show that $\sum_1^\infty{a_n^2}<\infty$ and that the reverse isnt true.
0
votes
0answers
19 views

Find the limit of :$\lim_{n\to\infty}n\sin(2\pi en!)$ [duplicate]

Find the limit: $\lim_{n\to\infty}n\sin(2\pi en!)$
1
vote
1answer
30 views

convergence of $\sum_{n=1}^\infty \int_0^{\frac {1}{n}} \dfrac {\sqrt x}{1+x^2} dx$

Test convergence of $\sum_{n=1}^\infty \int_0^{\frac {1}{n}} \dfrac {\sqrt x}{1+x^2} dx$ Attempt: Since, $\lim_{n \rightarrow \infty} \dfrac {1}{n} \in (0,1) \implies \sum_{n=1}^\infty \int_0^{\frac ...
5
votes
0answers
48 views

$\zeta(2)$ Euler's proof (Basel problem) [duplicate]

At one point Euler assumes that $$\frac{\sin x}{x} = \prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi} \right)\left(1-\frac{-x}{n\pi} \right)$$ Why does he assume that? If we factor random functions ...
5
votes
1answer
96 views

An alternating series identity with a hidden hyperbolic tangent

Is there an elementary (or at least a not-too-hard...) way to prove the following identity? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi - 5\log(2))$$ The identity follows ...
6
votes
1answer
51 views

How to prove $\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$

I am trying to prove the following: $$\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$$ I tried some values and it seems convincing. I wonder if this is a ...
0
votes
1answer
47 views

Convergence of $\sum_{n=1}^\infty \frac{\cos(3^nx)}{2^n}$

For what values of $x$ does $$\sum_{n=1}^\infty \frac{\cos(3^nx)}{2^n}$$ converge? I tried as follows: Since $\frac{\cos (3^nx)}{2^n} \le \frac{1}{2^n}$ for all $n$, therefore $\sum \frac{\cos ...