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

learn more… | top users | synonyms (5)

1
vote
2answers
112 views

Proof of convergence of $\sum\limits_{n=1}^\infty \frac{(2n)!!}{(2n+3)!!} $

I am interested in proofs of the convergence of the following series: $$\sum\limits_{n=1}^\infty \frac{(2n)!!}{(2n+3)!!} $$ I know it converges using Raabe's test. However, I am not allowed to use ...
2
votes
2answers
72 views

a problem on arithmetic progression,it is a very confusing sum

In an arithmetic progression,the sum of five terms is equal to 1\4 of the next five terms,prove that the 20th term is -112?
3
votes
3answers
179 views

Infinite Series $1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$

Was given the following infinite sum in class as a question, while we were talking about Taylor series expansions of $\ln(1+x)$ and $\arctan(x)$: $$1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$$ ...
5
votes
1answer
70 views

Combinatorics question - count how many ways to write $1,2,…,n$ with a certain order

A valid sequence is a sequence of length $n$ from the numbers $1,2,...,n$ such that: 1) every number appears once 2) apart from the first number in the sequence, every number $k$ has either a $k-1$ ...
0
votes
1answer
60 views

determine the convergence region of a complex series

Determine the region $\Omega$ of the complex plane such that for any $z\in\Omega$ the following series converges: $\sum_{n=1}^\infty\frac{1}{n^2}\exp(\frac{nz}{z-2})$. I do not know how to treat ...
0
votes
2answers
91 views

Construct a sequence such that the associated series converges to $\pi$

I have to construct a sequence $(x_n)$ such that $\sum_{n=0}^∞ x_n =\pi$. Is it valid to say $x_0=\pi$ and $x_n=0$ for all $n>0$ ?
6
votes
2answers
85 views

Calculate with sequences from OEIS

Is there any easy way to do calculations with sequences from OEIS online? For example I would like to input something like: (A007620(n+1) / 2 ) + A000027(n) and ...
1
vote
2answers
221 views

Information about Riemann Zeta function

I have general question on Riemann Zeta function. How can I improve knowledge on Riemann Zeta Function theory up to research? For example , what are the best books on Zeta ...
4
votes
1answer
46 views

$f^{n_i}(x)\to y$ implies $f^{-n_i}(y)\to x$?

Let $(X, d)$ be a compact metric space and $f:X\to X$ be a homeomorphism. If there exists a sequence $n_i$ such that $n_i\to\infty$ as $i\to\infty$ and $x, y\in X$ are such that $f^{n_i}(x)\to y$ as ...
9
votes
1answer
272 views

Intuitively, why is the Euler-Mascheroni constant near sqrt(1/3)?

Questions that ask for "intuitive" reasons are admittedly subjective, but I suspect some people will find this interesting. Some time ago, I was struck by the coincidence that the Euler-Mascheroni ...
2
votes
2answers
85 views

if $G_{4n+1}=2G_{2n+1}-G_{n},G_{4n+3}=3G_{2n+1}-2G_{n}$,Find $G_{n}=n?$

let sequence $\{G_{n}\}$ such $$G_{1}=1,G_{3}=3,G_{2n}=G_{n}$$ $$G_{4n+1}=2G_{2n+1}-G_{n},G_{4n+3}=3G_{2n+1}-2G_{n}$$ If such $G_{n}=n$, then we said $n$ is 'good'. How many 'good' numbers ...
8
votes
3answers
112 views

Proving that $\lim_{n \rightarrow\infty} \int_{0}^{\frac{\pi}{2}} \sin(t^n) dt =0$

It is clear that $\lim_{n \rightarrow\infty} \int_{0}^{1} \sin(t^n) dt =0$. (which is not what is to be proved here) I don't know how to proceed with the remaining part of the integral ie $\lim_{n ...
4
votes
3answers
74 views

sequence defined by $u_0=1/2$ and the recurrence relation $u_{n+1}=1-u_n^2$

I want to study the sequence defined by $u_0=1/2$ and the recurrence relation $$u_{n+1}=1-u_n^2\qquad n\ge0.$$ I calculated sufficient terms to understand that this sequence does not converge because ...
1
vote
1answer
72 views

max {$f_n(x):x\in[a,b]$}$\to$ max{$f(x):x\in[a,b]$}

Let $f_n:[a,b]\to\mathbb R$ be a sequence of continuous function converging uniformly to $f$. Show that $(1)$ max {$f_n(x):x\in[a,b]$}$\to$ max{$f(x):x\in[a,b]$} $(2)$ The above result does ...
0
votes
1answer
64 views

proof that $\sum_{i=0}^{\infty}\frac{1}{i!}$ converges to the mathematical constant or exponential constant $e$. [duplicate]

So basically my approach was to try to transform the lim definition of e to a sum but I didn't succeed in my difficult approach, so what is your way to prove ...
1
vote
1answer
44 views

Inserting means between 2 numbers?

Let $A_1, A_2, A_3... A_{2012}$ and $H_1, H_2, H_3 ... H_{2012}$ be arithmetic and harmonic means between $a$ and $b$ respectively. If $A_{1006}H_{1007}$ = 2013 then what is: ...
1
vote
0answers
67 views

$\prod_{n}f_{n}$ converges uniformly $\Rightarrow $ $\sum_{n}\mathrm{Log}(f_{n})$ converges uniformly

Let $\prod_{n}f_{n}$ be an infinite product of holomorphic functions on a given domain $\Omega$ converging uniformly on compact subsets of $\Omega$ to $f$. Then is it true that ...
1
vote
2answers
152 views

Showing summation is bounded

I'm currently taking a Comp Sci class that is reviewing Calculus 2. I have a question: Show that the summation $\sum_{i=1}^{n}\frac{1}{i^2}$ is bounded above by a constant I realize that this ...
1
vote
0answers
58 views

Limit of a quotient involving the gamma function

This is a continuation of another question I asked. It seems only incidentally related to statistics, so I figured it would be better separate. (Also, I'm not able to make nearly as much progress on ...
7
votes
1answer
376 views

Convergence of series involving in iterated logarithms $\sum \frac{1}{n(\log n)^{\alpha_1}\cdots (\log^k(n))^{\alpha_k} }$

What is the quickest way to show when $$ S(\alpha_1,\alpha_2,\cdots,\alpha_k) = \sum\limits_{n=3}^\infty \frac{1}{n (\log n)^{\alpha_1}\cdots (\log^k(n))^{\alpha_k}} $$ converges, where $\log^k(n)$ ...
1
vote
1answer
57 views

$\frac{f'(z)}{f(z)}= \sum_{n=1}^{+ \infty}\frac{f'_{n}(z)}{f_{n}(z)}$

I 've found this exercise. Let $\{f_{n}\}$ be a sequence of holomorphic functions on a given domain $\Omega$. Suppose that $\prod_{1}^{\infty}f_{n}$ converges uniformly on compact subsets of $\Omega$ ...
0
votes
0answers
33 views

Limit of Series with Variable Lower Bound [duplicate]

I'm trying to compute the following limit of a series: $$\lim_{n\to\infty} \sum_{k = n+1}^{\infty}\frac{e^{-n}n^{k}}{k!}$$ Factoring $e^{-n}$ out of the sum, applying the definition of $e^{x}$ as a ...
0
votes
2answers
70 views

Proof of $\lim_{n\to\infty}\left(\frac{a_n+b_n}{2}\right)^n=\sqrt{ab}$

Let $a_n$ and $b_n$ two strictly positive sequences such that $$\lim_{n\to\infty}a_n^n=a>0\qquad \lim_{n\to\infty}b_n^n=b>0.$$ I need to prove that ...
0
votes
1answer
42 views

Prove what the sum of $\sum\limits_{n=1}^\infty f(n)-f(n-1)$ is

$\sum\limits_{n=1}^\infty f(n)-f(n-1)$ I have three series that look like the above, is there a general way to prove this? I have one function where $f(n) = \sin \frac{-2}{n}$ and $f(n+1) = \sin ...
0
votes
3answers
43 views

Struggling to find the formula

I've been given a set of numbers (13 in total) and have been asked to find the formula. I'm normally pretty good at maths, but this has got me stumped. Can anyone either find the formula to work out ...
1
vote
1answer
108 views

Sum of polynomial

If $p(x)$ is a polynomial of degree $m$, does the polynomial $q(x)$ of degree $m+1$ exist so that $\sum_{i=0}^{n}p(i)=q(n)$? And if so, is there an algorithm to find the expression for $q(x)$?
1
vote
1answer
39 views

Converging sequence with sum

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and $x_1 \in \mathbb{R}^n$. Consider the sequence $x_{k+1} := f(x_k)$, for $k \geq 1$, such that $\lim_{k \rightarrow \infty} x_k = ...
1
vote
2answers
62 views

Maclaurin series for $f(x)=\frac{1}{1+x+x^2} $

What is the Maclaurin expansion of $f(x)=\dfrac{1}{1+x+x^2} $? Thank you! Edit: By multiplying both terms with $ (1-x) $ I got to $\dfrac{1}{1-x^3}-\dfrac{x}{1-x^3}$. Is it correct to transform ...
2
votes
1answer
45 views

An Euler-Mascheroni-like sequence [duplicate]

How does one compute the limit of the sequence: $$\sum_{k = 0}^{n}\frac{1}{3k+1} - \frac{\ln(n)}{3}$$ I would apreciate a hint.
2
votes
2answers
66 views

Difficulties to understand series

Hi in my book there is a series: $$\sum_{n=1}^{\infty} (-1)^n \cdot \frac{1}{n}$$ the context is convergence and so on. And this series serves as an exampel for a not absolut convergent but ...
6
votes
1answer
279 views

How find this sum $\sum_{k=1}^{n}\frac{(a_{k}+1)(b_{k}+1)}{a_{k}+b_{k}+3}$

let sequence $\{a_{n}\},\{b_{n}\}$ such $$a_{1}=2,b_{1}=1$$ and $$a_{n+1}=\dfrac{a_{n}+1}{a_{n}+b_{n}+3},b_{n+1}=\dfrac{b_{n}+1}{a_{n}+b_{n}+3}$$ find the ...
0
votes
2answers
66 views

Few calculus questions

$\lim \dfrac{(-1+i\sqrt{3})^{3n+1}}{8^n-1}$ I have no idea how to find limits of complex numbers. $\sum \dfrac{(n!)^2}{2^{n^{2}}}x^n, x \in$ C I have no idea how to find limits of complex ...
2
votes
4answers
141 views

Sum $\sum_{k=0}^{2013}2^ka_{k}$

let real sequence $a_{0},a_{1},a_{2},\cdots,a_{n}$,such $$a_{0}=2013,a_{n}=-\dfrac{2013}{n}\sum_{k=0}^{n-1}a_{k},n\ge 1$$ How find this sum $$\sum_{k=0}^{2013}2^ka_{k}$$ My idea: since ...
0
votes
1answer
26 views

Proving this question

If first two terms of a positive AP and a positive GP are equal then prove that an AP terms can not be greater than the corresponding GP term.I assumed AP and a GP but then getting no idea ahead.
0
votes
1answer
55 views

Does the series $\sum_{n=1}^\infty \frac {x^n} {a^n-b^n}$ converge?

Suppose there are the series $\displaystyle\sum_{n=1}^\infty \dfrac {x^n} {a^n-b^n}$, $\,a>b>0$. I tried to calculate radius of convergence by $\dfrac1R=\limsup\sqrt[n]{\dfrac 1 {a^n-b^n}}$ but ...
1
vote
1answer
59 views

How to prove: If $a \to -\infty $ and $b$ is bounded from below by a constant $k\in\Bbb R^{>0}$, then the $a\cdot b\to -\infty$

I must proof the following, with $a: \Bbb{N} \to \Bbb{R}$ and $b: \Bbb{N} \to \Bbb{R}$ If $a \to -\infty\ (n\to\infty)$ and $b$ is bounded from below by a constant $k\in\mathbb R^{>0}$, then the ...
0
votes
1answer
30 views

Proof-checking: $a \to +\infty \wedge \exists k \in \Bbb{R}(\forall r \in \Bbb{N}(k \leq b(r))) \Rightarrow (a+ b) \to+\infty$

let be $a: \Bbb{N} \to \Bbb{R}$, and $b: \Bbb{N} \to \Bbb{R}$, I must proof the following: "$a \longrightarrow +\infty \wedge \exists k \in \Bbb{R}(\forall r \in \Bbb{N}(k \leq b(r))) \Rightarrow ...
1
vote
1answer
1k views

How to test convergence of complex series?

I've been looking for examples of how complex series are tested on convergence, however I could not quite find what I wanted. So I'm asking here, how do I handle, for example: $$ \sum_{n=1}^{\infty} ...
20
votes
2answers
493 views

How to either prove or disprove if it is possible to arrange a series of numbers such the sum of any two adjacent number adds up to a prime number

I'm wondering if it's possible to write a theorem to prove or disprove the possibility of arranging a sequence of numbers (1,2,...n) such that the sum of any two numbers adds up to a prime number. An ...
2
votes
1answer
140 views

Mysterious subleading corrections to sums with internal dependence on limit

Is there a standard method for finding expansions in $N$ of sums like $$S(N)=\sum_{n=0}^N \sqrt{N^2-n^2}$$ beyond the first term? It is easy to compute here that $$S(N)=N^2 \int_0 ^1 \sqrt{1-x^2} ...
1
vote
1answer
96 views

Null sequence monotone decreasing.

Let $g$ be a rational function in $n$ variables $x_1,...,x_n$. Let $f_k$ be an exponential, logarithmic or power function for $k=1,...,n$. Let $f(x)=g(f_1(x),...,f_n(x))$ , and define a sequence by: ...
0
votes
2answers
64 views

Shifting Method

I'm taking a Course in Computer Science where we're having a refresher on Calculus 2 material. There is a problem that I don't understand or know how to do. Compute the following using the ...
4
votes
2answers
75 views

How to show that the limit of this sequence is $L=4$ (ex.8.11 Mathematical Analysis 2nd ed.- Apostol)

I need to show that this sequence has limit $L=4$. I know it could be useful the principle of mathematical induction but I can't understand the way I should use this principle to prove the limit is ...
3
votes
3answers
115 views

Sequence is periodic $x_{n+2}=|x_{n+1}-x_{n-1}|$

How to show that the sequence $$x_n, n \geq 0, x_{n+2}=|x_{n+1}-x_{n-1}|, n \geq 1$$ with $x_0, x_1, x_2$ positive integers, not all null, is periodic? I tried to pick up the square but obtained ...
0
votes
1answer
242 views

$\inf{(A+B)}=\inf{A}+\inf{B}$: An $\epsilon$ Proof

$$\text{COMPARATIVE EXAMPLE}$$ So I've been told that in order to show that $$\sup{(A+B)}=\sup{A}+\sup{B}$$ for non-empty and bounded above sets $A,B\subseteq\mathbb{R}$ one must show that ...
4
votes
0answers
59 views

Closest Pair between 2 sets of powers

Assume $a$ and $b$ are natural numbers, $A=\{a,a^2,a^3,\cdots\}$ and $B=\{b,b^2,b^3,\cdots\}$, find $\min\left|a_i-b_j\right|$ where $a_i\in A$ and $b_j\in B$. For example, if $a=3$, $b=10$, then ...
0
votes
1answer
63 views

Prove sequence inequality using Mean Value Theorem

Let $${a_n} = \sum\limits_{j = 1}^n {\frac{1}{j}} - \log n$$. Prove:$$\frac{1}{{n + 1}} - \frac{1}{n} < {a_{n + 1}} - {a_n} < 0$$ I was guided to use Lagrange's MVT. At first, It ...
2
votes
1answer
54 views

How prove this $\frac{1}{b_{k+1}b_{k+2}}+\frac{1}{b_{k+2}b_{k+3}}+\cdots+\frac{1}{b_{2k}b_{2k+1}}>\frac{1}{12345},$

let sequence $\{b_{n}\}$,and $b_{n}>0$,let $$S_{n}=b_{1}+b_{2}+\cdots+b_{n}\le n^{\frac{3}{2}},\forall n\ge 1$$ show that ...
0
votes
2answers
59 views

Series equivalent to differences of inverse primes?

Would someone be kind enough to correct my error here? $S=1+2+3+4+\cdots$ Now starting with the 1st prime number, regroup the sum: $S=(1+3+5+7+\cdots) + (2+4+6+\cdots)$ $S=(1+3+5+7+\cdots) + ...
0
votes
1answer
43 views

$e_{n+1} = K e_n e_{n-1} $ is $|e_{n+1}| = C|e_n|^{\varphi}$?

if $ e_{n+1} = K e_n e_{n-1} $ ($K$ is a constant, and $e_n$ is a serise), then, $ | e_{n+1} | = C|e_n|^\varphi$($C$ is constant) and $\varphi$ is golden ratio. Is this true? If yes, How can I show ...