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

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0
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0answers
7 views

$Find first positive perfect square in polynomial time

I have a quadratic. for example $$1x^2+6884x+3297$$ Is it possible to find the first perfect square in the series in polynomial time where both x and y are whole positive integers. In the above ...
4
votes
2answers
44 views

Limiting value of $\frac{x^n e^x}{n!}$ as $n\to\infty$

For the Taylor Series the remainder is of the form $$R_n = \frac{(x-a)^n}{n!} f^{(n)}(\xi) $$ with $a \leq \xi \leq x$ For the series of $e^x$ about $0$ (that is, the Maclaurin series) the remainder ...
0
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0answers
11 views

Does this property of time dependent sequences have a name?

For $i\in \mathbb{N}$ let $ \chi_i \colon \mathbb{R}^+ \to \mathbb{R}$ be such that for each $t\in \mathbb{R}^+$ we have $\chi_i(t) \in \mathcal{l}_2$ Suppose further that ...
8
votes
2answers
303 views

Can we add an uncountable number of elements, and can this sum be finite?

Can we add an uncountable number of elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. Any help ...
5
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1answer
24 views

Proving that and how $ \frac{1}{n}\sum\limits_{k\le n}\lfloor n/p_k\rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
1
vote
1answer
35 views

limit of sum $\dfrac{(-1)^{n-1}}{2^{2n-1}}$

What is: $$\sum^{\infty}_{n=1}\dfrac{(-1)^{n-1}}{2^{2n-1}}$$ I have done a Leibniz convergence test and proved that this series converges, but I do not know how to find the limit. Any suggestions?
2
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2answers
35 views

What function is this? $\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$ [on hold]

I am interested to know what function represents the following series: $$\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$$
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2answers
27 views

Method of finite differences solutions help?

For homework we were given this sequence: -2 8 27 85 260 ____ 2365 And asked to find the number in the blank. Well, I got the ...
0
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2answers
31 views

Sequences and Series - Find the value of n for which…

I am having some difficulty trying to solve this question. I have been given this question - Find the correct value of the letter n for which Xn = 5n - 2 and ...
3
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0answers
34 views

Is there anything known about the zeros of $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$?

Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that: $$f(s) :=\displaystyle \sum_{n=1}^{\infty} ...
3
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3answers
91 views

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$?

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$? It is well-known that $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k^2}=\frac{\pi^2}{6}$, so ...
5
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2answers
118 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$$ $B_n$ are the Bernoulli numbers. Mathematica does not help.
0
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1answer
49 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$ [duplicate]

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$$ Mathematica does not help.
1
vote
1answer
101 views

Solve this integral:$\int_0^\infty\dfrac{\arctan x}{x(x^2+1)}\mathrm dx$

I occasionally found that $\displaystyle\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\dfrac{\pi}{2}\ln 2$. I tried that $$\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\int_0^{\frac{\pi}{2}}x \ \mathrm ...
2
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1answer
53 views

Algorithm for computing square roots.

Fix a positive number $\alpha$. Choose $x_1>\sqrt{\alpha}$ and define $x_2, x_3, x_4, \dots$ by the recursion formula $$x_{n+1}=\frac{1}{2}\left(x_n+\frac{\alpha}{x_n}\right).$$It's easy to check ...
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1answer
33 views

infinite series question involving sigma

how to find out the sum of infinite series question $$\displaystyle\mathop{\sum^{\infty}\sum^{\infty}\sum^{\infty}}_{i=0\ j=0\ k=0\ i\neq j\neq k}\frac{1}{3^i}\cdot \frac{1}{3^j}\cdot \frac{1}{3^k}$$ ...
1
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0answers
55 views

Limit of recursive sequence $x_{n+1}=\frac1n(x_1+2x_2+3x_3+…+nx_n)$

I was trying to solve the following limit but I just can't get it: Let $x_1 = a$, $a>0$, and, for every $n \in \mathbb{N}$, $$x_{n+1}=\frac{x_1+2x_2+3x_3+...+nx_n}{n}.$$ Determine : ...
1
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1answer
40 views

Interchanging the order of summation for a particular double series.

I suspect, based on numerical approximation, that $$\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{(-1)^m}{2m+1}\frac{1}{n^{2m+1}} = ...
0
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4answers
72 views

Convergence or divergence of the series $\sum\limits_{n = 1}^{\infty} \sin(\pi/n)$

Let $ u_{n} = \sin \! \left( \dfrac{\pi}{n} \right) $, where $ n \in \Bbb{N} $, and consider the series $ \displaystyle \sum_{n = 1}^{\infty} u_{n} $. Which of the following is/are true? (a) $ ...
0
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2answers
43 views

Can absolute value functions be moved like this?

If I have an expression that looks like $|x-a_1| + |x-a_2| + |x-a_3| + ... + |x-a_n|$ Is it the same as doing $|nx - \sum_{i=1}^{n}a_i|$
1
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1answer
26 views

Identical Summation and Integration of specific functions

A strange coincidence which I discovered recently, is that $$\int_{-\infty}^{\infty}{\tan^{-1}{\frac{1}{(x-\alpha)^2+\frac{3}{4}}}}dx = ...
1
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1answer
59 views

What is the limit of the sum of “last half” part of harmonic series?

I'm looking for the limit of this sum: $\frac{1}{\left\lceil\frac{n}{2}\right\rceil+1}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+2}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+3}+\cdots+\frac{1}{n}$ ...
11
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3answers
149 views

Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing

I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing. $\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for ...
1
vote
2answers
52 views

Limit of a partial sum [on hold]

I want to find the limit $$\lim_{n\to \infty}\sum_{i=1}^n \frac{1}{n+i}$$ I tried this. But I am not able to do it. Can anyone please help how to proceed?
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0answers
25 views

Name for kind of big O notation with leading coefficient

Context: As known the big O notation $O(f(n))$ describes a function $g(n)$ such that there is a constant $C \ge 0$ with $\limsup_{n\to\infty} \left|\frac{g(n)}{f(n)}\right| \le C$ (I assume that ...
1
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1answer
32 views

How many ways can an integer $i$ appearing in a sequence with multiplicty at least $j$, be minimal

Let us construct an integer sequence of length $n$, where the integers are chosen from $\{1, 2, ..., k\}$, with i.i.d. uniform probability $\frac{1}{k}$. I want to compute the probability ($p_{ij}$) ...
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0answers
17 views

Comparison of bivariate generating functions

Suppose we have two bivariate ordinary generating functions describing two integer sequences which have indicies $a,b$ and $c, d$ respectively. Is there a straightforward way to determine, from ...
2
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2answers
25 views

A analytic representation of q- rational series

Using Mathematica, we can find $$\sum\limits_{n = 1}^\infty {\frac{{{{\left( {1 - q} \right)}^2}{q^n}}}{{\left( {1 - {q^n}} \right)\left( {1 - {q^{n + 1}}} \right)}}} = q,\;q \in \left( {0,1} ...
1
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0answers
41 views

Problem 14 from Baby Rudin chapter 3

Let $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1},\quad a_n=s_n-s_{n-1} \quad\text{for} \quad n\geqslant 1$$ Assume ...
26
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6answers
2k views

A way to calculate e?

Define three sequences: The first sequence is $$n^n: 1,\ 4,\ 27,\ 256,\ 3125,\ 46656, \ldots$$ The second sequence is that of the ratios between adjacent members of the first series, or ...
1
vote
1answer
142 views

Why is $1+2+3+\cdots = 0 $? [duplicate]

I had seen this result a while back in a Numberphile video: $1+2+3+\cdots = -\frac{1}{12}$ I was trying to prove the same result using a different method when I accidently proved that the sum was ...
2
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0answers
34 views

convergence of general harmonic series

My question is about determining the convergence of a general harmonic series using the integral test. According to the following resource: pg.32 , we can see that for a general harmonic series ...
2
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6answers
114 views

Find the limit of the sequence $(4^n)/((2n)!)$

How to find $$\displaystyle \lim_{n\to\infty} \frac{4^n}{(2n)!}$$ Can I use l'Hopital's rule?
1
vote
1answer
25 views

Dirichlet theorem

Can anyone give a simple number theory proof for the Dirichlet theorem? Statement of Dirichlet theorem:given any two numbers a and b whose g.c.d is 1,Prove that infinitely many primes exist in the ...
2
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2answers
59 views

How can I solve $\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$ step by step? [on hold]

$$\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$$ First off I know the answer to this question, but I do not know how to actually get the answer I do know some basic sums like that one. Can someone ...
0
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1answer
67 views

Prove sequence ${x_{n + 1}} = \sin {x_n},{x_1} = 1 $ has a limit

Prove that the sequence defined by $${x_{n + 1}} = \sin {x_n},\ {x_1} = 1$$ has a limit. Ok, I want to prove by Weierstrass: This sequence is monotonically decreasing Sequence is bounded ...
0
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1answer
55 views

high order (infinite series)

This question, I have made but there was no answer, so I will try again. If we have the sums $f(n) = 1^{59} + 2^{59} + 3^{59} + \cdots + (10^n)^{59}$ and $g(n) = 1^{5} + 2^{5} + 3^{5} + \cdots + ...
3
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2answers
47 views

Calculating in closed form another digamma alternating series

Is there any clever way of finish it fastly? $$\sum _{n=1}^{\infty } (-1)^{n+1} \left(\psi ^{(0)}\left(\frac{5}{8}+\frac{3 n}{8}\right)-\psi ^{(0)}\left(\frac{1}{8}+\frac{3 n}{8}\right)\right)$$ ...
1
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2answers
54 views

Explicit formula for recursive sequence.

Consider the series defined by $$P_n=(P_{n-1}-a)b\ .$$ $$P_0=c$$ Basically the number sequence $P_n$ represents the current Principal balance of a debt and constants $a$ and $b$ are constants or ...
0
votes
0answers
31 views

Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ with given conditions

If for any series $b(n)=2b(n-1)$ when $b(n)$ is odd number and $b(n)=b(n-1)$ if $b(n)$ is even number. then Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ Our Approach: I could not ...
1
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3answers
58 views

Why does this sum converge $\sum\limits_{k=1}^\infty\left (\frac{k\sin k}{2k+1}\right)^k$

I don't understand why this sum converges. $$\sum\limits_{k=1}^\infty \left(\frac{k\sin k}{2k+1}\right)^k$$ $$\lim_{x\to\infty} \left(\frac{k\sin k}{2k+1}\right) = diverge$$ I don't find any other ...
1
vote
1answer
67 views

Closed form of this sum

$$\sum _{ s=1 }^{ \infty }{ \left( \frac { 1 }{ 4s-1 } \sum _{ n=0 }^{ \infty }{ \left( \frac { 1 }{ n+1 } \sum _{ k=0 }^{ n }{ \left( \left( \begin{matrix} n \\ k \end{matrix} \right) \frac { { ...
4
votes
3answers
69 views

Sum involving zeta functions

Find closed form of the following - $$ \displaystyle \sum_{n=2}^{\infty}{\left(\frac{(n-1)\zeta(n)}{4n-1}\right)} $$ I don't know how to approach to it - Using the integral definition? I cannot use ...
0
votes
1answer
14 views

Summatory problem | Ordinary least square estimator

How I can transform the first expression in the second? \begin{align} \hat{\beta}_{1} & =\frac{n\sum X_{i}Y_{i}-\sum X_{i}\sum Y_{i}}{n\sum X_{i}^{2}-\left(\sum X_{i}\right)^{2}} \\ & = ...
0
votes
1answer
18 views

Piecewise $C_1$ and piecewise continuous

I would appreciate if the following questions could be clarified with your help. If a function is piecewise $C_1$, does this imply that it's also piecewise continuous? If a function is piecewise ...
0
votes
1answer
16 views

$f_n(x) = \left\lfloor \frac{\sin(2\pi (x / n + 1/ 4) + 1 }{2}\right\rfloor$ and related

$f_n(x) = \left\lfloor \frac{ \sin(2\pi (\frac{x}{n} + \frac{1}{4})) + 1}{2}\right \rfloor = 1 \iff x = kn$ and $ f_n(x) = 0 \iff x \neq kn$. Let $g_n(x)$ be what's within the floor brackets. Then ...
0
votes
0answers
35 views

sequence and their arithmetic means

Can it happen that $s_n>0$ and that $\limsup s_n=\infty$, although $\lim \sigma_n=0$ where $\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1}$. I want to take such sequence: $s_n$ is $\frac{1}{n}$ if $n$ is ...
1
vote
2answers
35 views

Cauchy's root test for series divergence

Just a question regarding determining the divergent in this example :$$\sum{ 1 \over \sqrt {n(n+1)}} $$ is divergent. It explains the reason by saying that $a_n$ > $1 \over n+1$. If I am not wrong it ...
3
votes
3answers
60 views

Calculate $\lim (\frac{1}{{1\cdot2}} + \frac{1}{{2\cdot3}} + \frac{1}{{3\cdot4}} + \cdots + \frac{1}{{n(n + 1)}})$

Calculate $$\lim \left(\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \cdots + \frac{1}{n(n + 1)}\right) $$ If reduce to a common denominator we get: $$\lim \left(\frac{X}{{n!(n + ...
-2
votes
2answers
42 views

Prove that if $\lim_{n\to \infty} a_n=l$ then $\lim \sup a_n=\lim \inf a_n=l$

Let $(a_n)_{n\in\mathbb{N}}\subset\mathbb{R}$, a bounded sequence. For each $n\in\mathbb{N}$, we have $A_n=\{a_k:k\ge n\}$. Let $\lambda_n=\sup A_n$ and $\beta_n=\inf A_n$.So we have $(\lambda_n)$ and ...