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

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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
488 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) ...
34
votes
6answers
756 views
0
votes
2answers
47 views

How to conjecture a formula of a sequence

I am trying to conjecture a formula for the $n$th term of $\{a_n\}$ if the first ten terms of the sequence are as follows 1.) $$2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366$$ 2.) $$1, 1, 0, 1, ...
1
vote
3answers
67 views

Is $(n^{1/n}-1)\in O(n^{-\frac12})$ as $n\to \infty$?

Let $$x_n=n^{1/n}-1$$ An exercise asks me to prove that $\{x_n\}\to 0$ and that $x_n\in O(n^{-\frac12})$ as $n\to \infty$. Now, I could easily prove the first part. But to me $x_n\notin ...
2
votes
1answer
45 views

Finding a general term of a series beginning with one

I'm trying to find a general term for this series: $$1 + \frac{x}{1\cdot 2} + \frac{x^2}{2\cdot 3} + \frac{x^3}{3\cdot 4} + ...$$ Without the one it's straightforward: $$\frac{x^n}{n(n+1)}$$ ...
0
votes
1answer
19 views

Proof limit of ratio of sequence .

Prove that as $n\to\infty$ $$\frac{1}{x_n} \to \frac{1}{x}$$ where we are also given $x_n \to x$, and $x_n,x\neq0$ Attempt: Suppose $x_n \to x$. Then for every $\epsilon > 0$, there exists a ...
0
votes
1answer
20 views

Square root of Sequence approaches square root value.

Suppose that $x$ is a real number, and $x_n\geq 0$, and $x_n→x$ as n grows. Prove that $\sqrt {x_n}→\sqrt x$ as $n$ grows. Attempt: Case 1: $x = 0$. Suppose that $x$ is a real number, and $x_n \geq ...
0
votes
3answers
56 views

How to show that $\sum_{n=1}^{\infty} \frac{1}{n^k}$ converges for all integer $k > 1$?

How to show that $$\sum_{n=1}^{\infty} \frac{1}{n^k}$$ converges for all integer $k > 1$? I know that comparison test would suffice to show that, but don't know how to start.
1
vote
1answer
51 views

Attempt to prove that every real number is a limit of a sequence of rational numbers

Prove that given a real number $x$, there exists a rational sequence $r_n$ such that $r_n \to x$ as $n$ grows. Proof: Suppose $x$ is a real number. Then we know by definition, there exists a ...
2
votes
1answer
44 views

Prove that $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have the same limit

I'm trying to solve the following problem prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit. In this post http://math.stackexchange.com/a/267499, I do not understand the following ...
3
votes
3answers
114 views

Summing infinitely many numbers: how to assign a value?

If we take $S = 1-1+1-1+1-1+1-1+...$ we can show (in many different ways) that the result of the sum is $\frac{1}{2}$. One way for example would be to add $S$ to itself but shift it along one place, ...
1
vote
1answer
44 views

Infinite Product implies divergence or not?

If $\displaystyle\prod_{n=1}^{\infty} (1-a_{n}) = 0$ then is it always true that $\displaystyle\sum_{n=1}^{\infty} a_{n} $ diverges? ($0 \leq a_{n} < 1) $
1
vote
3answers
55 views

Find $a_1$ given that $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$

If $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$, then $a_1$ is .. The options are $1$, $2$, $99$ or $100$. I'm sure the problem is trivial, but I just don't understand what is meant.
1
vote
1answer
23 views

Sequence diverging to infinitive.

Prove that the sequence diverges to negative infinitive. $ x_n = n - 3n^2$ as $n$ grows. Proof: By definition, a sequence approaches negative infinitive iff for every real number $M$, there exists a ...
0
votes
0answers
16 views

Recursively defined sequences and Explicitly defined sequences

Let $x_n$ be a real number sequence. I believe boundedness for $x_n $ is defined as : $x_n$ is bounded above iff $\exists c \in R, \forall n \in N , x_n \leq c $ . $x_n$ is bounded ...
2
votes
1answer
53 views

How can I compute the following fast?

What approach should I adopt for computing the following problem fast? $$f(n) = \sum_{i=1}^n (n \mod i)$$ $$ 1\le n \le 10^{10}$$ Since the answer can be huge I have to output it modulo some given ...
1
vote
0answers
37 views

Expectation of $\frac{1}{X+1}$ for a geometric random variable

I am confused over $E(\frac{1}{1+X})$ where $X$ is geometric distribution with parameter $p$. The book wants me to prove that $E(\frac{1}{1+X})=log((1-p)^{\frac{p}{p-1}})$ Here's what I did. ...
0
votes
0answers
66 views

Limit of sequence $(1+\frac{1}{n^2})^n$ [closed]

Obviously I know that $(1+\frac{1}{n})^n=e$ but can anyone help with how to prove that $\lim{(1+\frac{1}{n^2})^n}=1$ using the $N, \epsilon$ definition? My instinct is to use the binomial theorem, ...
-1
votes
0answers
18 views

Is there any way to solve the following series? [closed]

Is there any solution of this series ? Any solution would greatly help.
1
vote
2answers
44 views

Closed form sum for the series given below?

Does the following series have a closed form sum? $$f(n,r) = \sum_{i=0}^n \binom{r+i}{r}$$
5
votes
1answer
127 views

Computing $\int_0^{\large1/1^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\int_0^{\large 1/2^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\cdots $

What tools would you recommend me for computing this series? $$\int_0^{\large1/1^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\int_0^{\large 1/2^2} \left\{ \frac{1}{t} ...
1
vote
3answers
39 views

Why is $\sum\limits_{i=0}^{n}(r-1)^{n-i}{n\choose i} = r^n$?

I was solving a problem and found that $\sum\limits_{i=0}^{n}2^{n-i}{n\choose i} = 3^n$. So I tried to generalise it and got $\sum\limits_{i=0}^{n}(r-1)^{n-i}{n\choose i} = r^n$. Is it true for $r ...
4
votes
9answers
411 views

Prove $ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $

$$ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $$ Wolframalpha shows that it is a correct identity, ...
0
votes
0answers
21 views

hyperbolic series $\sum_{r=1}^n \cosh(rx)$

I have attempted to do this question on hyperbolic functions: Prove that $$\cosh x + \cosh 2x + ... + \cosh nx ...
11
votes
1answer
269 views

Another Ramanujan's formula dealing with $\coth^{2}(5\pi)$

Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = ...
0
votes
0answers
10 views

What series describes the feedback of a fully connected network with signal strength at each node converging to 1?

Take a fully connected network with $N$ nodes operating in lockstep. One and only one node will receive a signal from an external source at a time step $t$, but the node receiving it is random. The ...
2
votes
3answers
118 views

Monotonicity of the sequences $\left(1+\frac1n\right)^n$, $\left(1-\frac1n\right)^n$ and $\left(1+\frac1n\right)^{n+1}$

I am working on the following sequences. $$x_n=\left(1+\frac1n\right)^n \qquad z_n=\left(1-\frac1n\right)^n \qquad y_n=\left(1+\frac1n\right)^{n+1}$$ I am trying to prove that $x_n$ and $z_n$ are ...
2
votes
1answer
27 views

$\sum_{n=1}^{\infty} {(1-\cos(\sin 1/n))}^{w}$ with $w$ as parameter

Let $f(x)=(1-\cos(\sin x))$; $a_n=f(1/n)$ for $n\in\mathbb{N}$ For which $w>0$ series $$\sum_{n=1}^{\infty} {a_n}^{w}$$ converge? I haven't got a slicest idea how to check that, absolutely none ...
2
votes
1answer
43 views

Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation \begin{equation} \sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2} \end{equation} defined for $0\le x\le 2\pi$. Here is ...
1
vote
0answers
40 views

What is the most “powerfull” method to prove a sequence is increasing or decreasing?

Given a sequence $a_n$ defined in a recursive manner, the methods I know to prove if the sequence is increasing are: 1) observe if $a_{n+1} - a_n > 0 \ \forall n.$ 2) take $\frac{a_{n+1}}{a_n}$ ...
3
votes
1answer
46 views

Sign of a series

Someone could compute the sign of the following series ? \begin{equation} \underset{k > 0}{\sum} \frac{\sin (kx)}{k} \end{equation} I expect that is the same as the first term $\sin x$ because of ...
-1
votes
0answers
43 views

How to find the result of infinite series $\sum\limits_{n=1}^\infty\frac{(-1)^{n+1}}{n^2}$? [duplicate]

As per wolfram-alpha the result is $\frac{\pi^2}{12}$. How to calculate it manually? $$\sum\limits_{n=1}^\infty\frac{(-1)^{n+1}}{n^2}=\frac{\pi^2}{12}$$
1
vote
1answer
21 views

How can I construct a specific sigmoid function?

The simple sigmoid function $$f(x)=1/(1+e^{−x})$$ approaches zero as x tends to negative infinity, and approaches $1$ as x tends to positive infinity. But I want to set $1$ and $20$ instead of $0$ and ...
-3
votes
0answers
21 views

Is the series convergent or divergent [closed]

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum \sqrt{a_na_{n+1}}$ always convergent?Either prove it or give a counterexample.
26
votes
1answer
1k views

Is $\sum_{n=1}^\infty \frac{\sin(2n)}{1+\cos^4(n)}$ convergent?

As I was just checking this 'child prodigy' out on Youtube, I stumbled upon this video, in which Glenn Beck asks the kid to do the following proof: Further on, the kid starts sketching a proof ...
0
votes
1answer
290 views

provide a combinatorial proof that $C_{n+1} = C_0C_n + C_1C_{n-1} + …. + C_kC_{n-k} + …C_nC_0$

(a) Let $C_n$ denote the number of ways of writing a valid list of open and closed parentheses of length $2n$ (valid means that at any point along the list, the number of open parentheses must be ...
0
votes
2answers
28 views

Sequence converging to zero

Prove the sequence $x_n = \frac{2n}{n^2 + 1}$ converges to zero. Attempt proof: $x_n = \frac{2n}{n^2\left(1 + \frac1{n^2}\right)} = \frac2{n\left( 1 + \frac1{n^2}\right)}$ Now we can know $\frac2n ...
1
vote
2answers
46 views

on the exercise 8.10 Apostol. (limit of sequence)

The exercise states: prove that the limit of the sequence $$a_{n+2}=(a_na_{n+1})^{1/2} \ where \ a_1 \ge 0, a_2 \ge 0 $$ is $L = (a_1a_2^2)^{1/3}$ The solutin says: $$Let \ b_n = ...
0
votes
1answer
26 views

Assumption in a convergence proof

Im in the middle of a proof of the fact that for a>0, if lim $x_n$ = a, then lim $\sqrt{x_n}$ = $\sqrt{a}$. I'm in the step that i use $| \sqrt{x_n} - \sqrt{a} |$ = $ \frac{|x_n - a|}{ ...
6
votes
2answers
106 views

Convergence of $S_n(a) = \sum_{an < k \leqslant (a+1)n} \frac{1}{\sqrt{kn-an^2}}$

Let $a\in\mathbb{R}$ et $n \in\mathbb{N}$, Denote the following sequence, $$\displaystyle S_n(a) = \sum_{an < k \leqslant (a+1)n} \frac{1}{\sqrt{kn-an^2}}$$ For which values ​​of $a$ the ...
2
votes
0answers
25 views

Evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ where $n,k$ are fixed

Is there a general way/technique to evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ in terms of $r$, where we consider $n$ and $k$ fixed natural numbers and $n > k$? (here, ...
4
votes
3answers
169 views

Proving that the following series is convergent

Can someone please help me prove that this series is convergent? $$ \sum_{i=1}^\infty \left({n^2+1\over n^2+n+1}\right)^{n^2} $$ I guess I'm supposed to show that the limit of the sequence is an "e" ...
2
votes
2answers
78 views

Proving the convergence of $a_{n+2}=(a_{n+1}a_n)^{1/2} \qquad (a_1\ge0, a_2\ge0)$

I am trying to solve exercise 8.10 from Apostol-Mathematical Analysis. I need to prove that the following sequence converges to $L=(a_1a_2^2)^{1/3}$. $$a_{n+2}=(a_{n+1}a_n)^{1/2} \qquad (a_1\ge0, ...
0
votes
1answer
30 views

Rules for constructing a sequence (induction)

What steps are required to built a sequence? Here is an example: Show that there is a sequence of rational numbers that converge to pi. Note that $0 < \pi$. Since the rationals are dense in the ...
7
votes
0answers
96 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
0
votes
1answer
20 views

Lim sup ratio test and Ramifications

Let $x_n$ be a real number sequence. I have managed to prove that,for a sequence $x_n$ of positive terms : if lim sup $(x_{n+1}/x_{n}) < 1$ then $x_n$ is not only eventually monotonicly ...
0
votes
0answers
16 views

Relation between monotonocity and Lim sup of Ratio

Let $x_n$ be a real number sequence. Is it true that : i) if $x_n$ is a bounded eventually monotonicly non-increasing sequence, then lim sup $(x_{n+1}/x_{n}) \leq 1 $. ii) If $x_n$ is a ...
0
votes
1answer
33 views

Arithmetic sequence: determine number of addends

Given that the sum of first 5 terms of an arithmetic sequence is 65, the sum of the last five terms is 1090,and the sum of all the terms is 5313.Find the number of terms in the sequence I try to ...