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

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-1
votes
1answer
42 views

absolutely convergent & conditionally convergent [closed]

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
81 views

Explain about absolute convergence and convergence [closed]

$$\sum_{n=1}^{\infty}\frac{{{(-1)^{n}\sin (n)}}}{{{n}}} $$ Explain about absolutely convergent and convergent. Thx a lot.
1
vote
0answers
23 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
59 views

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

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

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

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
23 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 [closed]

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
24 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
49 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
30 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 ...
6
votes
3answers
170 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 to obtain 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 ...
1
vote
2answers
52 views

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

Find the sum of $\sum \limits_{n=1}^{\infty} \frac{2^{-n}(n+3)}{(n+1)(n+2)}$
1
vote
1answer
59 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
25 views

Recurrence relations and initial conditions [closed]

I couldn't figure out how to do the super/subscript, hence the photo.
-2
votes
1answer
41 views

How to find the formula of S3n? [closed]

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? 4 first terms is like natural number, but at ...
4
votes
2answers
54 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
52 views

Problem with sequences and series problems [closed]

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
19 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
80 views

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

Find all natural sequences for which holds $$a_n=a_{a_{n-1}}+a_{a_{n+1}}$$ a) for all natural numbers $n\ge 2$ b) for all natural numbers $n\ge 3$ I tried to do something with the characteristic ...
1
vote
1answer
92 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 ...
1
vote
1answer
27 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
72 views

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

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
39 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 ...
6
votes
2answers
225 views
+50

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi - 5\log(2))$$ The identity follows ...
6
votes
1answer
54 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
48 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 ...
3
votes
1answer
69 views

How do I find this limit as n goes to infinity $(1 - \frac {1}{3} ) (1 - \frac {1}{6} ) (1 - \frac {1}{10} )$ … $(1 - \frac {1}{n ( n+1) /2} )$

How do I find this limit as n goes to infinity $(1 - \frac {1}{3} ) (1 - \frac {1}{6} ) (1 - \frac {1}{10} )$ ... (1 - $\frac{1}{\frac12 n ( n+1)} )$
0
votes
3answers
21 views

Convergence divergence of one series from another

Let ${a_n}$ be sequence of real numbers. Which of following is true: If $\sum a_n $converges then $ \sum a_n^4$ converges If $\sum |a_n| $ converges then $\sum a_n^2$ converges If $\sum a_n $ ...
0
votes
0answers
10 views

The expected number of mutations in a sequence of elements, each with random delays

In a sequence, the number of the permutations, is the (minimum) number of the pair of elements needed to switch to make them sorted. For example in the following: ...
-2
votes
3answers
41 views

Finding the height of a bouncing ball. (Using the geometric series in an applied setting)

I am doing a problem in a textbook (Boas' Mathematical Methods in the Physical Sciences) where a ball is dropped from a height of one yard and the sum of vertical distance in each drop is the series: ...
1
vote
3answers
65 views

Proof that limit as n approaches inf of $3^n \cdot \frac{1}{n!} = 0$ using squeeze theorem

This is a homework question. I'm completely new to epsilon proofs, so I'm pretty bad at them. I want to prove that $3^n \cdot \frac{1}{(n!)} = 0$ converges to 0. Here's where I'm at. By the ...
0
votes
0answers
23 views

Formula for two sequences

I have two sequences, a list of $x$'s and a list of $y$'s. From $x[0]$, i need to derive $y[0]$ ... so $x[0]$ (some math here) $= y[0]$ $x[1]$ (some math here) $= y[1]$ ... ... $x[n]$ (some math ...
-2
votes
1answer
52 views

How to prove $\lim_{x\to 0}\sum_{n=0}^{\infty}\frac{1-3\ x}{(1+n\ x)^5}$ is not convergent [closed]

Nowadays I encounter a limit which is difficult for me to evaluate it. Please help me to evaluate it. Thank you. $$\lim_{x\to 0}\sum_{n=0}^{\infty}\frac{1-3\ x}{(1+n\ x)^5}$$
0
votes
0answers
13 views

K-Uniformity of Infinite Sequence

A book on random number generators refers to the subject of infinite-uniform infinite sequences as being "random." I was wondering if anyone could shed light on the definition of K-Uniform Infinite ...
-2
votes
1answer
66 views

How to compute $\sum_{n\geq 0}\frac{\sin n}{n!}$?

I want to calculate the sum of $$\sum_{n\geq 0}\frac{\sin n}{n!}.$$ I think I am supposed to use the Taylor polynomial of $\ e^x$ but I don't know how to solve it. Thanks for your help.
1
vote
1answer
86 views

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

Test Convergence of $\displaystyle\sum_{n=2}^\infty \dfrac {1}{(\log n )^3}$ Attempt: I haven't been able to find a suitable comparator for $\dfrac {1}{(\log n )^3}$ . The integration test also seems ...
2
votes
4answers
78 views

Evaluate $\lim_{n \rightarrow \infty} \frac {[(n+1)(n+2)\cdots(n+n)]^{1/n}}{n}$

Evaluate $$\lim_{n \rightarrow \infty~} \dfrac {[(n+1)(n+2)\cdots(n+n)]^{\dfrac {1}{n}}}{n}$$ Attempt: Let $$y=\lim_{n \rightarrow \infty} \dfrac {[(n+1)(n+2)\cdots(n+n)]^{\dfrac {1}{n}}}{n}$$ ...
2
votes
2answers
95 views

Finding limit using Riemann integral

$$\lim _{n\rightarrow \infty }\sum_{i=1}^{n}\frac{n}{\left ( i-1 \right )^{2}+n^{2}}$$ What is the idea behind this? I have watched an MIT open courseware video on this kind of problems, and what I ...
2
votes
1answer
49 views

tricky question in combinatorics - deck of cards [closed]

A deck of cards with $4$ sets, each set contains $13$ cards. We want to create a new sequence of $n$ cards: each time we choose a card, write it down as the next element in the sequence, put it back ...
1
vote
2answers
26 views

$\sum_1^\infty{\frac{(-1)^n}{\ln{(2\cosh{n})}}}$ convergence

How can I test this serie for convergence ? Logarithmic criterion didnt seem to work. How to test if it converges absolutely ? As Leibnitz works $$\sum_1^\infty{\frac{(-1)^n}{\ln{(2\cosh{n})}}}$$
2
votes
6answers
86 views

Convergence of $\sum\limits_{n=1}^\infty \frac {n+1}{2^n}$.

Test Convergence of $$\sum\limits_{n=1}^\infty \dfrac {n+1}{2^n}$$ Attempt: $$\sum\limits_{n=1}^\infty \dfrac {n+1}{2^n} = \sum\limits_{n=1}^\infty \dfrac {n }{2^n} + \sum\limits_{n=1}^\infty \dfrac ...
1
vote
1answer
26 views

2013th powered sequence

Let $a_1$, $a_2$, ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the 2013th power of the digits of $a_n$. Do there exist distinct positive ...
1
vote
1answer
36 views

Test convergence of $\sum_{n=2}^\infty{(\ln{n})^{-n}}$

$$\sum_{n=2}^\infty{(\ln{n})^{-n}}$$ How can I test convergence for this sum? I can get a conclusion with Ratio test but limit is hard. With Cauchy condension test I cant come to a conclusion.
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
24 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 ...
0
votes
1answer
36 views

Does $\Sigma_{n=1}^{\infty} \frac {\sqrt {2n-1}~ \log (4n+1)} {n(n+1)}$ 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 [closed]

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
127 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 ...