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

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0
votes
3answers
53 views

finding a confusing limit

yeah just another limit. I have $Xn=\dfrac{1000^n}{\sqrt{n!}} + 1$ that $+1$ confuses me any hints & solutions how to calculate limit will be apreciated
2
votes
1answer
2k views

The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
1
vote
2answers
60 views

Limsup, showing that two expressions are equal

I am stuck at this problem which I use for something else. If $\{a_i\}$ is a sequence of number then I want to prove that $\limsup |a_i|^{1/i}=\limsup|a_{i+k}|^{|1/i}$, where k is a fixed positive ...
0
votes
0answers
28 views

Multiplying double sums formula

Questions: 1)Can I modify the same-endpoint formula somehow to get different-endpoint formula? http://en.wikipedia.org/wiki/Cauchy_product#Finite_summations 2)Or even better do you know any formula ...
3
votes
1answer
67 views

Telescoping property: $\sum_{k \geq 0}\frac{1}{(4k+1)(4k+3)}$

I need to calculate the sum $$\sum_{k \geq 0}\frac{1}{(4k+1)(4k+3)}$$ I've made some attempts to transform this in a summation that I could apply the telescopic property, but I didnt have any ...
19
votes
5answers
468 views

Computing $\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$

What ways would you propose for the limit below? $$\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$$ Thanks in advance for your suggestions, hints! Sis.
0
votes
0answers
37 views

Hyperreal probability density?

I'm fairly new to the subject of hyperreal numbers and I'm wondering if there exists an infinitesimal number $a$ such that (in some reasonable sense) $$\sum_{n=1}^\infty a=1$$ ? In other words: Is ...
-3
votes
1answer
69 views

Prove $\displaystyle\sum_{k=0}^\infty \frac{2^k}{\binom{2k+1}{k}}=\frac{\pi}{2}$ [closed]

How to show that $$\sum_{k=0}^\infty \frac{2^k}{\binom{2k+1}{k}}=\frac{\pi}{2}$$
2
votes
0answers
32 views

Question about the value of the Infinite series [duplicate]

I tried all day long to get the value of the following infinite series.. $\displaystyle \sum_{k=1}^{\infty} \dfrac{2^{k}}{{2k+1}\choose {k}}$ which seems to be $\dfrac{\pi}{2}$ by my matlab ...
1
vote
2answers
44 views

Discrete Math sequence question

The question is find $a3$: $a_0 = 2, a_1 = 4$ and $a_{k+2} = 3a_{k+1}-a_k$ for any integer $k \geq 0 $ I know the answer is 26, although how do you get the answer?
1
vote
3answers
47 views

How to find $n$-th value in a series

Let $(x_n, y_n, z_n) = (3, 1, 0)$ for $n=0$ For $n \ge 1$, $$\begin{align} x_n &= x_{n-1} +3 z_{n-1}\\ y_n &= x_{n-1} +2 z_{n-1}\\ z_n &= 5 y_{n-1} \end{align}$$ Please let me know the ...
2
votes
2answers
236 views

A question on a proof that every sequence has a monotone subsequence

Consider the following proof that every sequence $\{x_n\}_n$ in $\mathbb R$ has a monotone subsequence. Proof: Let us call a positive integer $n$ a peak of the sequence if $m > n \implies x_n ...
1
vote
1answer
17 views

Compound interest problem with increasing deposits

An Investor starts with an initial investment : $A$ He earns a steady profit of 10 percent per year. But every year he adds additional amount which increases by 15 percent every year. At the end of ...
1
vote
1answer
36 views

Convergence of a sum of sines

If $ s_N(x) := \sum_{n = 1}^N c_n \sin(n x) $ converges uniformly on $[0, \pi]$ as $N \to \infty$ then $c_n = o(n^{-1})$. a) Is $c_n = o(n^{-1})$ sufficient for uniform convergence? b) Is $\sum_n n ...
7
votes
2answers
324 views

Limit: How to Conclude

I have difficulty to conclude this limit ....; place of my attempts and results, can anyone help? tanks in advance $$\lim_{x\to +a}\, \left(1+6\left(\frac{\sin ...
52
votes
11answers
2k views

Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums. ...
3
votes
1answer
56 views

Does the series $\sum_{i=0}^{\infty}\exp\{{-(m!)!}\}(D^m\phi)(0)$ converge for every $\phi \in C^\infty$?

Does the series $$\sum_{m=0}^{\infty}\exp\{{-(m!)!}\}(D^m\phi)(0)$$ converge for every $\phi \in C^\infty$? For analytic function $\phi$, we can show that the series converges by using ...
9
votes
2answers
136 views

Summation of series $\sum_{k=0}^\infty 2^k/\binom{2k+1}{k}$

How to find the sum of this series? $$\sum_{k=0}^{\infty}\cfrac{{2}^{k}}{\binom{2k+1}{k}}$$ It seems very easy. But I still can not work it out, can anyone help?
10
votes
5answers
750 views

$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$ Evaluate Integral

Anyone remember the method for this? I think this should been done on the site $$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$$
3
votes
4answers
191 views

What is $\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$?

I'm asked to find $\displaystyle\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$. It seems that the sum converges. I managed to prove that $\forall n,1<\sum_{k=1}^n ...
0
votes
0answers
17 views

Solving Power Series Equality

I'm trying to solve the following power series equation to get the coefficients $\{a_k\}$: $$\sum_{k=1} ^ \infty a_k x^k = \sum_{\ell = 0} ^\infty \frac{x}{\ell !} \left( \sum_{k=1} ^ \infty a_k x^k ...
3
votes
1answer
54 views

An equality from Representation Theory

Studying Representation Theory of finite groups I've bumped in the following identity: $$\frac{n(n+1)}{2}=\sum_{i=1}^n\frac{(2i-1)!!(2n-2i+1)!!}{(2i-2)!!(2n-2i)!!}$$ My book suggests to prove it ...
2
votes
1answer
41 views

Divergence of a series containing primes

Is there an easy proof showing that the series $1/p$, where $p$ changes over prime numbers, is divergent?
1
vote
1answer
49 views

Can I pick any N in the epsilon N - definition?

If I were asked to pick a $N$ that would satisfy the definition such that for any $n > N$, the distance between $f(n)$ and the limit would be smaller than a specific epsilon.... ...can I just pick ...
-5
votes
0answers
20 views

Prove that every bounded sequence has a convergent subsequence. [duplicate]

Let $(a_n)$ be a bounded sequence. Let $(a_{n_k})$ be a monotonic subsequence of $(a_n)$. Since $(a_n)$ is bounded $(a_{n_k})$ is also bounded. $(a_n)$ is bounded monotonic sequence and hence ...
1
vote
1answer
25 views

Finding limes superior and limes inferior

Find the limes superior and the limes inferior of the sequence $$ c_k=\frac{k+(-1)^k(2k+1)}{k}, k\in\mathbb{N}_+. $$ Do not know if there is a special way to find limit points... my ...
32
votes
0answers
613 views

Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?

Let $a_0=1,a_n=\tan{a_{n-1}}$. Then is $\{a_n\}_{n=0}^\infty$ dense in $\Bbb{R}$? I've drawn a map of this dynamical system and it seems that the sequence is dense on $\Bbb{R}$.
-3
votes
1answer
58 views

Consider $f_n(x)={x^n-x^{3n}}$

A. For what values of x is the function series is point-wise convergent, and to what function? B. For what values of x is the series uniform convergence? My answers in the textbook are: A. As $n\to ...
0
votes
1answer
17 views

Question about spectrum versus spectral sequences

What is the difference between spectrum sequences and spectral sequences? Are they considered to be the same? I know that the spectrum sequence of a real number $\alpha$ is the sequence that has ...
0
votes
0answers
22 views

Unique sum between elements of a numerical set

I require a set of numerical elements on wich the sum of some of these elements is unique to the set, that it's to say, no other combination in the sum of elements will result the same outcome. ...
0
votes
0answers
19 views

Proving a property of the largest limit point

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence. By Bolzano-Weierstraß this sequence does have a limit point. Let $\bar{a}$ denote the largest limit point of the sequence. Show that among ...
0
votes
3answers
37 views

Does the series $\sum (1+n^2)^{-1/4}$ converge or diverge?

The integral is $\int { (1+n^2)^{-1/4}}dn$ is not quite possible, so I should make a comparison test. What is your suggestion? EDIT: And what about the series $$\sum (1+n^2)^{-1/4} \cos ...
-1
votes
0answers
52 views

gradually changing sequence of integers [closed]

I would like to clarify a question regarding a sequence of numbers. For the sequence 8,5,3,2 , if I were to give the next 3 integers, should it be 2,3,5 or 2,1,1? Help is greatly appreciated.. What ...
1
vote
1answer
41 views

Will this series converge? If so, what is its limit?

If $a_n=(a_{n-1}+a_{n-2})/2$ and $a_1, a_2$ are given, will this series converge? And if so, what is the limit? By intuition I think it converges to $(a_1+2a_2)/3$ , but I am not able to prove it.
2
votes
1answer
23 views

Proving $\{x_n\}$ converges to $a$ when $|x_n-a|\le Cb_n$ for large $n$ and $C$ is a positive constant.

If $\{b_n\}$ is a sequence of nonnegative numbers that converges to $0$, and $\{x_n\}$ is a real sequence that satisfies $|x_n-a|\le Cb_n$ for large $n$, where $C$ is a fixed positive constant, prove ...
0
votes
0answers
54 views

Prove $x_n$ converges to $a$ iff every subsequence of $x_n$ also converges to $a$.

Prove $x_n$ converges to $a$ iff every subsequence of $x_n$ also converges to $a$. Suppose that $\{x_n\}$ is a sequence in $\mathbb R$. Definitions available: (1) A sequence of real numbers ...
2
votes
3answers
53 views

absolute convergence of an infinite series

Show that the series: $$1 + \frac{x}{1\cdot 2} + \frac{x^2}{2\cdot 3} + \frac{x^3}{3\cdot 4} + \cdots$$ is absolutely convergent when $-1< x <+1$. I've been trying to prove this however am ...
2
votes
1answer
19 views

solving Legendre equation using the Frobenius method around a singular point

My first question is whether I can solve the Legendre equation for $l=0$, i.e. $$ (1-x^2)y''(x)-2xy'(x)+y(x)=0, $$ using the regular power series method around $x=0$. My second question is, even if ...
1
vote
1answer
32 views

Which of the following series will converge and which one will diverge?

Can anyone help me out that which of the following series will converge and which one will diverge, with some explanation? A) $\sum_{n=1}^\infty \sin\left(\frac{\pi}n\right)$ B) $\sum_{n=1}^\infty ...
0
votes
3answers
34 views

Formula for the $n^{th}$ positive integer that is not divisible by 2 or 3.

The first few terms of this sequence are 1,5,7,11,13,17,... The numbers increase by alternating adding 4 or 2. From what I remember from Algebra II, since the second level of differences is constant, ...
0
votes
1answer
53 views

Closed form of the series

I want to evaluate $\sum_{i=1}^{n} (x+i)^4$ So what i did is, after expanding it i reduce it to following form $ x^{4} * n + 4 x^{3} * \sum_{i=1}^{n}i + 6x^2\sum_{i=1}^{n}i^{2} + ...
0
votes
3answers
25 views

how to expand this expression using series?

How does $$(R^2 +[|x|]^2)^{0.5}$$ expand as $$R + \frac{[|x|]^2}{2R} + \text{(more terms)}$$ around the point $x = 0$? I tried using a Taylor series but it didn't work out.
3
votes
1answer
30 views

Convergence of sequence of function

I need to check if sequence of functions $f_n(x):=\sqrt{x^2+\frac{1}{n}}$, $n\in \mathbb{N}$ converges (pointwisely, uniformly) in intervals:$[-1;1]$ and $\mathbb{R}$. Is there any algorithm how to ...
5
votes
3answers
681 views

Does the $\sum\limits_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$ converge?

Does the following series converge? $$\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$$ As $$\frac{1}{n\sqrt[n]{n}}=\frac{1}{n^{1+\frac{1}{n}}},$$ I was thinking that you may consider this as a p-series ...
12
votes
2answers
424 views

Compute $ \sum\limits_{m=1}^{\infty} \sum\limits_{n=1}^{\infty} \sum\limits_{p=1}^{\infty}\frac{(-1)^{m+n+p}}{m+n+p}$

How would you compute this sum? It's not a problem I need to immediately solve, but a problem that came to my mind today. I think that the generalization to more than three nested sums would be ...
0
votes
0answers
22 views

Defining multiplication of monotone increasing seuqences to be strictly monotone increasing.

I am constructing the reals as the set of equivalence classes of monotone increasing sequences of rationals. However, one problem I have is defining multiplication such that the product of two ...
2
votes
2answers
50 views

Convergence of $(a_n)$ when $(a_n^{1/n})$ converges

Let $(a_n)$ be a sequence of positive numbers such that the sequence $(a_n^{1/n})$ converges. What is a sufficient condition that $(a_n)$ also converges? $a_n=n$ is an example of a divergent sequence ...
3
votes
1answer
97 views

golden ratio from new formula? perhaps from theory of modular units?

Please consider the following infinite product series which I found by pure happenstance: $$\frac{1+\sqrt{5}}{2}= e^{\pi/6} \prod_{k=1}^\infty \frac{1+e^{-5(2k-1)\pi}}{1+e^{-(2k-1)\pi}}$$ My ...
5
votes
4answers
492 views

What explains this bizarre behavior?

Consider the sequence $x_{n+1} = 2x_{n}-\frac{1}{x_n},n\geq0 $. For $x_{0} = 0,87$ we have $$ \begin{aligned} X(1) &\approx 0,590574712643678 \\ X(2) &\approx -0,512116436915835\\ X(3) ...