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

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5
votes
1answer
71 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
1
vote
0answers
24 views

How prove this sequence $u_{m}=v_{m}$

Question: Assmue that $m$ is give postive integer numvers,define sequence $$\{u_{k}\},\{v_{k}\},u_{0}=v_{0}=u_{1}=v_{1}=1$$ and for any real numbers $a_{i},i=\{1,2,\cdots,m-1\}$, ...
1
vote
5answers
98 views

Sequence $\sin^2n$

I would like to prove that if $a_n = \sin^2(n),$ then it does not converge. Usually we show 2 subsequences with different limits for those sine cases, but I could not do this since my n is a natural ...
4
votes
2answers
44 views

$\lim_{N\to \infty} \sum_{n=N}^{2N} c_n = 0$ $\Rightarrow \sum_{n=1}^{\infty} c_n$ converges?

If $\lim_{N\to \infty} \sum_{n=N}^{2N} c_n = 0$ do we have that $\sum_{n=1}^{\infty} c_n$ converges? At first this did not seem true($\sum_{n=N}^{2N} (-1)^n$ is $0$ when N is odd), but I've failed to ...
1
vote
1answer
42 views

Two sequences defined by recurrence relations satisfy $x_n/y_n<\sqrt{7}$ for all $n$

Let $(x_n)_{n\geq 1}$ and $(y_n)_{n\geq 1}$ be two sequences such that: $$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$ with $x_1=2$ and $y_1=1$ Prove that for all $n$ ...
1
vote
0answers
46 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
-2
votes
1answer
336 views

geometric series word problem help [on hold]

Brennen has been playing a game where he can create towns and help his empire expand. each town he has allows him to create 1.15 times as many villagers. The game gave brennan five villagers to start ...
4
votes
6answers
250 views

Sum of an unorthodox infinite series

$ \frac{1}{2^1}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+\cdots $ This is a pretty unorthodox problem, and I'm not quite sure how to simplify it. Could I get a solution? Thanks.
7
votes
4answers
713 views

partial sum involving factorials

Here is an interesting series I ran across. It is a binomial-type identity. $\displaystyle \sum_{k=0}^{n}\frac{(2n-k)!\cdot 2^{k}}{(n-k)!}=4^{n}\cdot n!$ I tried all sorts of playing around, ...
5
votes
0answers
162 views

Sorting of prime gaps

Let $g_n $ be the $n^{th}$ prime gap $p_{n+1}-p_n.$ If we re-arrange the sequence $ (g_n)$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
3
votes
3answers
40 views

Convergence of $\sum \frac{1}{\ln n}(\sqrt{n+1} - \sqrt{n})$

Show that $$ \sum\frac{1}{\ln n}(\sqrt{n+1} - \sqrt{n})$$ Converges. I've tried the telescopic property or even write it as $$\sum \frac{1}{\ln n (\sqrt{n+1}+\sqrt{n})}$$ But didnt help. Thanks ...
5
votes
0answers
43 views

Convergence of infinite series of complex numbers [duplicate]

This has been bugging me for some months since our lecturer, a fields medalist, mentioned that he couldn't solve it when he was our age, yet had had two students submit solutions to it (during our ...
1
vote
3answers
41 views

How fast does a sequence with finite sum go to zero?

Suppose $z_n$ is a nonnegative sequence, monotonically decreasing to zero, and $$\sum_{i=1}^{\infty} z_i < 1.$$ Is it possible to translate this into a bound on the how small $z_i$ is? For example, ...
1
vote
2answers
27 views

Exact solution for rate parameter in finite geometric series

All, I'm working on a problem whereby, given a known quantity, Q, and number of intervals, n, I want to calculate the rate parameter -1 < r < 1 of a finite geometric series. Is there a way to ...
0
votes
1answer
17 views

A function relating $k$ and $j$, where $k=\max_{l\in \mathbb{N}}\sum_{i=0}^{l}2^{n-i}\leq j$ and $n= \lfloor \log_{2}j \rfloor$

Do you know any function that relates k and j, where $k=\max_{l\in \mathbb{N}}\sum_{i=0}^l 2^{n-i}\leq j$ and $n=\lfloor \log_2 j \rfloor$? So, say, for $j=3$: $n=1$ and $k=1$ because $3\geq ...
2
votes
1answer
75 views

Renormalizing Legendre polynomials to $P_n(0)=1$

One way to define the Legendre polynomials is with the recurrence relation $$(n+1)P_{n+1} (x) = (2n+1)xP_{n} (x)-nP_{n-1} (x),$$ with $P_0(x)=1$ and $P_1(x)=x$. This standardization is normalized so ...
8
votes
2answers
150 views

Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
0
votes
1answer
17 views

how many variables are there from 9 digits excluding repeat numbers

I have the numbers 1 to 9 I need to know how many different 9 digit code variations i would have using 1-9 but excluding any "next digit" replications. example: 123456789 is acceptable 112345678 ...
1
vote
1answer
51 views

Can I prove this, or hopeless? Deviating too much from mean

Can I prove this: We have a sequence of vectors $\left(X_i(n)\right)$ for $i=1,\ldots,t$, where $n\rightarrow \infty$. $t$ does depend on $n$ and is Chosen such that $1 \ll t \ll n$, for instance, ...
0
votes
1answer
24 views

Programming PARI/GP to do a sum

I'm trying to compute the following sum in PARI/GP $C=\sum_{n=1}^{\infty} \frac{g(n)}{n^2}$ where $g(n)$ defined as $$g(n)=(-1)^r, \qquad r=\text{number of even indexed prime factors of $n$}$$ By ...
0
votes
0answers
18 views

How to choose the parameter?

Can I choose Parameters $\beta \in (1,2)$ and $1 \ll x \ll n$, such that $$\sum_{i=1}^{x} i i^{-\beta} \gg \sum_{i=x+1}^{n} i i^{-\beta}$$ Would be great if you could give an example.. $\gg$ means ...
0
votes
0answers
17 views

Single formula sequence partitionining interval

So I have the real sequence for fixed $x\in \mathbb{R}$: $y_{j}(x)=\begin{cases}f_{j-1}(x) &\text{if } |f_{j-1}(x)|\leq |x_{j}|, \\ f_{j}(x) & \text{else}, \end{cases}$ where 1) ...
0
votes
4answers
32 views

If (m+n)th term of an Arithmetic Progression is p, and (m-n)th term is q, then find the mth and nth terms.

If (m+n)th term of an Arithmetic Progression is p, and (m-n)th term is q, then find the mth and nth terms.
3
votes
3answers
59 views

For an irrational number $a$ the fractional part of $na$ is dense in $[0,1]$

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
2
votes
1answer
24 views

Existence of a sequence of integers $\lbrace a_k\rbrace_{k\geq 1}$ so that the first $k$ digits of $a_k\alpha$ are $0$ where $\alpha$ is irrational.

Let $\alpha$ be an irrational number. Is there a sequence $\lbrace a_k\rbrace_{k\geq 1}$ of integers so that the first $k$ digits of the fractional part of $a_k·\alpha$ are $0$? (in base $2$, for ...
1
vote
1answer
29 views

Sequences and series

If $p, q, r$ are in G.P. and the equations: $$px^2 + 2qx + r = 0$$ $$dx^2 + 2ex + f = 0$$ Have a common root, then show that $$\frac{d}{p}, \frac{e}{q}, \frac{f}r$$ are in A.P. Well I tried taking ...
1
vote
1answer
46 views

4 crystal balls and a 10,000 story building

There is an analog of this question I've heard with 2 crystal balls but a higher number like 4 or more makes it much more interesting. You are given 4 crystal balls and there is a 10,000 story ...
3
votes
2answers
71 views

Closed form of a complex series sum

I am working on a proof that require a closed form (if that is not possible then at least a tight lower bound) of the expression below: $$A(n,k)=\sum_{i=1}^k ...
1
vote
1answer
69 views

interval of convergence of $e^x$

Can somebody explain how to find the interval of convergence for $$ e^x=\sum_{n=1}^\infty\frac{x^n}{n!} $$ I don't fully understand the ratio test/root test/integral test etc. and I don't understand ...
0
votes
0answers
30 views

Series representation for $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 \pi ^2 A^2+W^2}+6 \pi ^2 A^2+3 W^2}}{\sqrt{2}}$

My question is, is there a series representation or other function of $L$ and $A$ I can use when I solve the following equation for $W$? $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 ...
1
vote
2answers
36 views

Checking for convergence of series

To check the convergence of the series $\displaystyle \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+..\infty$ Attempt 1: Term $\displaystyle u_n= \frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$ ...
1
vote
1answer
43 views

An arctan criterion for convergence?

Is the following inference correct, and if so, is it a mere curiosity? Let $\{a_k\}_{k \in \Bbb N}$ be a sequence of positive real numbers and set $$ P_k = \sum_{j=0}^{k-1} a_k \quad \mbox{and} ...
4
votes
4answers
156 views

An asymptotic term for a finite sum involving Stirling numbers

The question is a by-product at the end of this post. The following asymptotic term will ensure the convergence of some series. $$ \frac{1}{n!} \sum_{k = 1 }^{n } \frac{{n \brack k}}{k+1} = ...
3
votes
0answers
56 views

Is there an easier way to find the “natural” integration constant?

Suppose we take consequtive derivatives of a function at a point and then interpolate them with Newton series (Newton interpolation formula) so to obtain a smooth curve. ...
0
votes
0answers
40 views

Infinite sum of all natural numbers and solution of infinite sum of positive odds and evens [on hold]

The infinite sum of all natural numbers is believed to be -1/12. Now if you have this sum and you rearrange it so all evens are together and all odds are together the value should still be -1/12. Now ...
7
votes
1answer
107 views

Showing that $\sum_{i=1}^n \frac{1}{|x-p_i|} \leq 8n \left( 1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} \right)$

I'm taking a summer analysis course and preparing for our final exam later this week. Our professor gave us the following problem on our mock exam, and I can't seem to get anywhere on it. Does anyone ...
15
votes
2answers
946 views

any pattern here ? (revised 2)

for any positive number $k$, I have a $(k+1)*(k+1)$ matrix. I wonder if these matrices follow any "obvious" pattern. My goal is to guess the elements for matrix with $k=5$ and above (most probably in ...
1
vote
3answers
59 views

Using Riemann sums to show that $\sum_{i=1}^n 1/i = \log{n} +c+O(1/n)$

I want to show that there exists a constant $c$ such that: $$ \sum_{i=1}^n 1/i = \log{n} +c+O(1/n) $$ I am thinking about Riemann sums. Any hints on that?
2
votes
3answers
70 views

Telescoping series: $\sum i^2 x^i$ for $0 < x < 1$

It is asked to find the sum $$ \sum_{i=1}^{\infty} i^2 x^i $$ Using the telescoping property. But I could not find a sequence to write my $s_n$ in function of and apply this.. Does anyone have a ...
0
votes
1answer
7 views

Sequence Interval of convergence

I could someone help me with the following sequence of functions of which I attempted to find the interval of convergence, but I couldn't get it to match with the solution I get from WolframAlpha ...
-1
votes
1answer
45 views

Find the next number in the series [on hold]

Can someone help me to find the next number of the following series 25,7,24,13,5,12,5,3,?
0
votes
1answer
68 views

Closed form of solution of recurrence equation

Does there exists a closed form of solution of the following recurrence equation: $$a_{n+1} =a_n^2 -a_n +1$$
1
vote
1answer
16 views

Making a sequence alternating

So I have the sequence $\{x_{j}=k+\frac{1+2(j-\sum_{i=0}^{k}2^{n-i})}{2^{n-k}}\}_{j\in \mathbb{N}}$, where $n= \lfloor log_{2}j \rfloor$ and k$=\{l\in \mathbb{N}:$maximum l s.t. ...
3
votes
1answer
57 views

How should I prove that: $\sum_{i=1} ^{n}(\sin(\frac{i\pi}{n}))^2=\frac{n}{2}$

$$\sum_{i=1} ^{n}\Big(\sin\big(\frac{i\pi}{n}\big)\Big)^2=\frac{n}{2}$$ An interesting conclusion and checked for validity...holds for $n\geq 2$, but yet do not know how to prove it. Are there any ...
4
votes
0answers
53 views

Integral's Closed-form expression in terms of hypergeometric function

I want to solve the following integral: $$I = 2\left[\int_{0}^{1}\dfrac{y^m}{(1 - ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y+\int_{0}^{1}\dfrac{y^m}{(1 + ay)^{m + 1}\sqrt{1 - y^2}}\mathrm{d}y\right]$$ ...
0
votes
1answer
32 views

how to prove using induction that sum of terms?

Prove that $\displaystyle\sum\limits_{i=1}^{k}\left(\dfrac{1}{(2i-1)}\dfrac{1}{(2i+1)}\right) = \dfrac{k}{(2k+1)}‎‎$ My Base of Induction is to check that it is true for i=1, so: ...
2
votes
1answer
229 views

Questions about finite sequences of natural numbers $(a_1, \dots, a_n)$ with distinct partial sums

I have a school assignment to do, but I have no idea, where to start. I hope you can help. Here it is: We have a finite sequence $A = (a_1, a_2,\ldots, a_n)$, length of $A$ is $n$, elements of $A$ ...
4
votes
6answers
762 views

Finding the sum to infinity

Question: Find the sum to infinity for the following series $$1, -\frac{1}{2}, \frac{1}{2^2}, -\frac{1}{2^3},\cdots$$ What would be the technique used to find such a sum?
1
vote
0answers
29 views

A question on infinite series and boundedness of sequence

Let $(a_n)$ be a real sequence such that for every convergent real series $\sum x_n$ of positive terms , $\sum |a_n|x_n$ is also convergent , then is it true that $(a_n)$ is a bounded sequence ?
2
votes
1answer
24 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...