Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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8
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1answer
108 views

Evaluate $\lim_{n \to \infty} (n!)^{\frac{1}{n^2}}$

Let $a_n = (n!)^{\frac{1}{n^2}}$. Now, $$n! \geq1 \implies (n!)^{\frac{1}{n^2}} \geq 1$$ and $$n! \leq n^n \implies (n!)^{\frac{1}{n^2}} \leq n^{\frac1n}$$ But $$\lim_{n \to \infty} n^{\frac1n} = 1 =...
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88 views

A relation concerning the “sum of squares” counting function $r_2(n)$

Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here the sign as well as order of summands matters. Also by convention we ...
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268 views

Interesting sequence involving prime numbers jumping on the number line.

Udpate 4: Trying to characterize finite and infinite cycles. Update 3: All primes $a_0\ge29$ seem to either have infinite cycles or finite non-terminating cycles that converge to infinite cycles of ...
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140 views

Does this “almost all integers in order” sequence have a closed form?

Can you help me define a formula for the following sequence (first $130$ terms) : ...
8
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1answer
432 views

Question B6 on Putnam 2016

I didn't make any headway on this during the Putnam, mainly because I wasn't going to waste my time on a B6 question which should be the most difficult one on the exam. It seems really intriguing, ...
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253 views

The Chudnovsky pi formula $1/\pi$ revisited

Define the constants, $$A=163\cdot1114806\\B=13591409\\C=640320$$ Given the binomial coefficient $\binom{n}{k}$, then we have the pi formulas, $$\frac{1}{\pi} =\frac{12}{(C)^{3/2}}\sum^\infty_{k=0} \...
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244 views

Modified AGM: $a_{n+1}=\frac{a_n+b_n}{2}, \quad b_{n+1}=a_n+b_n-\sqrt{a_n b_n}$

The idea is as follows: generate a sequence from two numbers by subtracting their means from their sum, for example arithmetic and geometric means: $$a_{n+1}=a_n+b_n-\frac{a_n+b_n}{2}=\frac{a_n+...
8
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333 views

A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$

This question is motivated by Why is $\pi$ so close to $3$?, Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$. I. $\pi$ and $\pi^2$ There are series with all terms positive for $\pi-...
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268 views

How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done with this series: $$ \sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$ As hardmath says in his comment, the series ...
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289 views

Name of a certain set

I want to know if there is any already-standard way to refer to the sets described as follows. For a set $X$, let $-X = \{-x: x \in X \}$; call it the negative of $X$. Take the set of all primes in $\...
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381 views

A sequence similar to the Catalan numbers

The $n$-th Catalan number $c_n$ has the closed form $\frac1{n+1}\binom{2n}{n}$ and follows the recursion $c_n = \sum\limits_{i = 0}^{n-1} c_{n-1-i}c_i$ I am interested in the quantity $e_n$ which ...
8
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1answer
258 views

Composition of Taylor Series

Suppose I have smooth functions $f,g,y_0$ and $y_1$ from $\mathbb{R}$ to $\mathbb{R}$, such that $$y_1(x) = y_0(x) - \epsilon g(y_0(x))$$ Then I consider $$f(y_0(x)) = f(y_1(x) + \epsilon g(y_0(x)))$$ ...
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135 views

The sine cardinal function and $F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = 0$

Define the function, $$F_n=\frac12-\int_0^\infty \frac{\sin^n x}{x^n}\,dx+\sum_{x=1}^\infty \frac{\sin^n x}{x^n}\tag1$$ where $\rm{sinc}^n(x)=\frac{\sin^n x}{x^n}$ is the sine cardinal function. We ...
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85 views

Analogous “Dark Sector” Trigonometric (and Hyperbolic) Functions

Existing Definitions: $$\zeta(n)=\sum_{k=1}^\infty \frac{ 1 }{k^n}$$ $$\lambda(n)=\sum_{k=1}^\infty \frac{ 1 }{(2k-1)^n}=\frac{\left(2^n-1\right)}{2^n}\zeta (n)$$ $$\eta(n)=\sum_{k=1}^\infty \frac{(-1)...
7
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152 views

Recurrence $a_{n}=a_{\lfloor 2n/3\rfloor}+a_{\lfloor n/3\rfloor}$

I am considering the sequence $$a_n=a_{\lfloor 2n/3\rfloor}+a_{\lfloor n/3\rfloor}$$ with $a_0=1$, and I would like to calculate the limit $$\lim_{n\to\infty} \frac{a_n}{n}$$ I have seen this famous ...
7
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1answer
93 views

On $\int_0^\infty \frac{\exp(-x^2)}{1+x^2}dx=\frac{\pi e}2\text{erfc}(1)$

I was attempting to answer this question, but then I came across a question of my own involving my attempt. Task: Prove $$\int_0^\infty\frac{\exp(-x^2)}{1+x^2}\mathrm dx=\frac{\pi e}2\text{erfc}(1)$$ ...
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121 views

Gap in spiral sequence

OEIS sequence A272573 describes a sequence generated in the following way: Start a spiral of numbers on a hexagonal tiling, with the initial hexagon as a(1) = 1. a(n) is the smallest positive ...
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87 views

Determining finitude or infinitude from a simple geometric construction

Playing with a pencil on a checkered sheet I encountered this construction: 1) take a point $A$ on the grid and a point $B$ that is distant from $A$ $n=2,3,4...$ horizontal steps and $1$ vertical ...
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239 views

Check simple proof that $\lim\limits_{s\to0^+}\sum\limits_{n=1}^\infty(-1)^nf(n)^{-s}=-\frac12$ if $f>0$, $f''\le0$ and $f(+\infty)=+\infty$

$f:[1,+\infty)\to \mathbb{R}_+$ satisfies $\ f''\leq0,\ f(+\infty)=+\infty $. Prove $$\lim_{s\to0^+}\sum_{n=1}^\infty (-1)^n[f(n)]^{-s}=-\frac{1}{2}$$ Some of my friends showed me this question ...
7
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1answer
113 views

Find the largest $n \in \mathbb{N_+} $ such that $\{ (2+\sqrt 2)^n\} < \frac{7}{8}$, where $\{x\}$ denotes the fractional part of $x$.

Problem Find the largest $n \in \mathbb{N_+} $ such that $\{ \left(2+\sqrt{2}\right)^n\} < \dfrac{7}{8}$, where $\{x\}$ denotes the fractional part of $x.$ My Solution First, we can prove that $...
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202 views

Generating function of the sequence $\binom{2n}{n}^3H_n$

Generating functions of the sequences $\binom{2n}{n}^2H_n$ and $\binom{2n}{n}^2H_{2n}$, where $H_n$ is $n$-th harmonic number, are known in terms of elliptic integrals $$ \sum_{n=1}^\infty\binom{2n}{n}...
7
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1answer
135 views

Examine uniform convergence of the series $\sum_{n=1}^\infty \frac{x}{(1+(n-1)x)(1+nx)}$

Examine uniform convergence of the series $$\sum_{n=1}^\infty \frac{x}{(1+(n-1)x)(1+nx)}$$ on the intervals $[a,b]$ where $(0 < a < b)$ and $[0,b]$ where $(b>0)$ My attempt: (i) On ...
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2answers
369 views

On the converse of the $n$th term test

A student asked a very insightful question in my Calculus class this morning. I did not know how to answer him. (Admittedly, I am not an analyst by trade: once I passed my qualifiers I never looked ...
7
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1answer
83 views

Flipping odd number array by two numbers in series.

Suppose an array of length $n$(odd number) which is composed with only $0$. We start to ‘flip’( or ‘change’) numbers with following rules. [1, 1] → [0, 0] [0, 0] → [1, 1] [1, 0] or [0, 1] : do not ...
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435 views

Closed Form for this Taylor Series?

Does anybody know whether or not this sum has a closed form? $$f(x)=\sum_{n=0}^\infty \frac{x^n}{n!(2^n+1)}$$ I can't get WA to even understand it when I type it in. For context, the reason I want to ...
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156 views

A sufficient condition for a sequence to converge if arithmetic mean of the sequence converges?

We have a well-known conclusion: If a sequence $\{a_n\}_{n\in\mathbb{N}}$ converges, then the arithmetic mean $\frac{S_n}{n}$ (where $S_n=\sum\limits_{k=1}^na_k$ is the nth partial sum) converges to ...
7
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1answer
117 views

Euler Identity Derivation-Is this allowed?

I was reading about how Euler derived his famous identity, $e^{i{\pi}}$. It said that it was discovered when Euler took the Taylor Expansion for $e^x$, and he multiplied the $x$ by $i$, and it gave ...
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269 views

What's infinte sum of the reciprocal of the primorial?

$$\sum_{n=1}^\infty \frac{1}{p_n\#} = \frac{1}{2}+\frac{1}{2\times3}+\frac{1}{2\times3\times5}+\dots$$ where $p_n\#$ is the nth Primorial. Does this sum approaches some known value or constant and ...
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65 views

Smallest number $N$, such that $\sqrt{N}-\lfloor\sqrt{N}\rfloor$ has a given continued fraction sequence

How can I find the smallest positive integer $N$, such that the continued fraction of $\sqrt{N}-\lfloor\sqrt{N}\rfloor$ begins with a given finite sequence containing a zero followed by positive ...
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0answers
134 views

Closed form for $\prod_{i=2}^{\infty} (1 - \frac{1}{i!})$

Question. I wonder whether there exists a closed form for the following infinite product $$ \prod_{i=2}^{\infty} (1 - \frac{1}{i!}) $$ I can prove that the product is convergent, but failed to ...
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337 views

Closed form of $\sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!}$

Is it possible to calculate the sum $$ \sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!} $$ in closed form? Formal naive argument gives $$ \sum _{n=0}^{\infty} \...
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142 views

Is $\sum_{n=1}^\infty \ln \left(1-\frac{1}{p_n^s} \right)=-\ln \zeta(s)$ the only known series with primes and nontrivial closed form?

To clarify, I'm asking about series where $n$th term depends only on $n$th prime number. From the famous result (by Euler, I think) we have: $$\sum_{n=1}^\infty \ln \left(1-\frac{1}{p_n^s} \right)=-\...
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224 views

For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

Unless I'm making a mistake, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely then the ...
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292 views

Is there a way to write this recurrence relation in faster-to-program manner?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
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666 views

Intuition about the Bernstein polynomials proof of the Weierstrass approximation theorem

The Weierstrass approximation theorem can be stated as follows: Let $f\in C([a,b])$, then there exists a sequence $(p_n)_{n\in \mathbb{N}}$ of polynomials in $[a,b]$ such that $(p_n)$ converges ...
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189 views

Finite sum with three binomial coefficients

I need to find a closed form expression, if there is one, of the following sum: $$\sum_{j=0}^m{n+1-k\choose j}{k-1\choose m-j}{A+2-k+m-j\choose m-j+2}$$ where all parameters are integers, $~1\leq k\...
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168 views

Evaluate $\sum\limits_{n=0}^\infty\frac{4^n}{(2^{2^{n}}+1)^2}$

I have known that $\displaystyle\sum_{n=0}^\infty\dfrac{2^n}{x^{2^{n}}+1}=\dfrac{1}{x-1}$ for $x>1$. Taking the derivative of both sides , $\displaystyle\sum_{n=0}^\infty\dfrac{4^nx^{2^n-1}}{(x^{...
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161 views

Double sum involving $\cos$

I ran across a double sum and was wondering if someone may be adept at evaluating it. I must admit that my double summation skills could be better, and I am always ready to learn more. Show that: $$\...
7
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1answer
206 views

Another conditionl leading to irrationality of $\sum _{k=1}^ \infty \dfrac 1{n_k}$?

If $\{n_k\}$ is a strictly increasing sequence of positive integers such that $\lim \inf _{k \to \infty} n_k ^{1/2^k} >1$ and $\lim _{k \to \infty} n_k^{1/2^k}$ does not exist , then is it true ...
7
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1answer
126 views

Infinite Integration in Limits of Integration

Given the following: $$ u_0 = \int \limits_{ 0 } ^{ 1 } x \, dx , \:\:\: u_1 = \int \limits^{ \int \limits_{ 1/2 } ^{ 1 } x \, dx } _{ \int \limits_{ 0 } ^{ 1/2 } x \, dx } x \,dx , \:\:\: u_2 = \int \...
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306 views

How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$?

How do I evaluate this sum :$$\displaystyle\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$$ Note : I used many criterions of convergence to show if it converges but i didn't up. Thank you for any ...
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178 views

Infinite series involving factorials of squares

Does $$\sum_{n=0}^\infty \frac{1}{(n^2)!}=2.04167\dots$$ possess a closed form?
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127 views

Minimizing beginning terms of a Fibonacci-like sequence

For $ a, b \in \mathbb{Z}^+ $ such that $ a = b $ or $ 2a < b $, let $ f_{a, b} : \mathbb{Z}^+ \to \mathbb{Z}^+ $ be a sequence that satisfies the following three criteria: $$ f(1) = a, \\ f(2) = ...
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84 views

Proving another digammabinomial series result

This series is related to some extent to the previous question of mine, that is Computing $\sum_{n=1}^{\infty} \frac{\psi\left(\frac{n+1}{2}\right)}{ \binom{2n}{n}}$, where an approach by series only ...
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164 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 ...
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799 views

Sum of infinite series defined recursively

Suppose $s$, $t$, and $\delta$ are constants satisfying $0<s<t<1$ and $\delta>1$. An infinite sequence $\{y_k\}_{k=1}^{\infty}$ defined as follows: The initial term, $y_1$, is the ...
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792 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
7
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0answers
404 views

Partial summation of a harmonic prime square series (Prime zeta functions)

I am trying to find the following series: $S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$ ...
7
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2answers
216 views

Factorizing the Sum of Two Fibonacci numbers

The Fibonacci and Lucas numbers are defined for all integers $n$ by the recurrence relations $$F_n=F_{n-1}+F_{n-2}\text{ where }F_1=1\text{ and }F_2=1,$$ $$L_n=L_{n-1}+L_{n-2}\text{ where }L_1=1\text{ ...
7
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1answer
158 views

Does there exist infinite words using the alphabet $\{A,B,C,D\}$ that avoids patterns $XX,\ XAX,\ XBX,\ XCX,\ XDX$?

Another form of this question is: Does there exist a gap-1 square-free infinite word using the alphabet {A,B,C,D}? Normally square-free in this context means that there are no sub-words twice in a ...