For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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2
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0answers
17 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
0
votes
0answers
10 views

Limit of a sequence with binomial coefficient. Can I use Stirling?

I was trying to solve this limit but I didn't reach to solve it: $\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $ What is the best way to solve it? Could I use Stirling relation? If no, why?
0
votes
2answers
12 views

Calculating limit of a series of series

Given is the following series I want to calculate the limit for $n \to \infty$. I already recognised the geometric series $\sum_{i=2}^n a^{i} = \frac{1}{1-a}$ for $a=e^\rho$ (since rho is ...
0
votes
0answers
11 views

What kind of information does the derivative of a sequence of functions give us?

When we derivate a function we know for example when then function increases and drecreases but when I derivate a sequence of functions I don't how to interpret the derivative since it is a collection ...
1
vote
2answers
19 views

Alternative version of definition of convergence

Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$. Consider a new ...
1
vote
1answer
67 views

Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$

Having $f(z)=\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n$ I had to find what $\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n\sum\limits_{n=0}^{+\infty} \frac{D_n}{n!}z^n=\sum\limits_{n=0}^{+\infty} ...
0
votes
2answers
16 views

How do I use the ratio test to determine convergence or divergence in this problem?

I have the problem: $$a_{n} = \frac{e^{n+5}}{\sqrt{n+7}(n+3)!}$$ I am told to use the ratio test to determine convergence or divergence (or the test could be inconclusive). So I take the limit: ...
1
vote
3answers
50 views

Solve the following sequence problem

Let a sequence be defined as $$a_n=\lim_ {x \to 0}{1-\cos (x)\cos(2x).....\cos (nx)\over x^2}$$ a)prove that the given sequence is monotonic and that it is not bounded above. b)calculate $$\lim_{n \to ...
0
votes
0answers
4 views

Lagarias and Robin theorems versus multiplicative property

If I use for example Robin's theorem, see here in the section Growth of arithmetic functions, or Lagarias equivalence, see (5) here has sense ask us what is the more sharp inequality for ...
-1
votes
2answers
38 views

$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$ iff $\lim_{n\to\infty}{(a_{n}-b_{n})=0}$ [on hold]

I need to proof or disproof that: $$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$$if and only if $$\lim_{n\to\infty}{(a_{n}-b_{n})=0}$$
-1
votes
0answers
14 views

Express $\prod \frac{a_i}{x+b_i}$ in terms of known functions

I am interested if there is a way to express the finite product $$ \prod_{i=1}^{S} \frac{a_i}{x+b_i} $$ in terms of known functions like Gamma, Beta, etc.
0
votes
1answer
21 views

Why is the following reverse triangle inequality true for given series?

I wish to show that for $(a_k)$ a sequence of numbers, $a_k \in \mathbb{R}$ then claim : $|\sum\limits_{k = n+1}^m a_k | \leq ||\sum\limits_{k = n+1}^\infty a_k| - |\sum\limits_{k = ...
4
votes
0answers
31 views

Variation of the Kempner series

It is easy to argue that the Kempner series converges : $$ \sum\limits_{\substack{n \text{ : 9 is}\\\text{ not a digit of } n}} \frac{1}{n} < \infty$$ Let $E \subset \Bbb N_{>0}$ the subset ...
3
votes
2answers
17 views

Arithmetic Sequences - accountant problem.

I need help with $part$ $c$ of the following problem: An accountant has a salary scheme as outlined below. STARTING SALARY £16,000. ANNUAL INCREASES OF £1,000 GUARANTEED THEREAFTER. Let £$u_n$ ...
0
votes
0answers
29 views

How can I change a summation of cosines to a product of cosines for higher degree functions?

I was wondering if I can get an alternate form of a sum over cosines $$\sum_{n=1}^m\cos{f(n)}$$ and I found that I can. We must however make a modification to the upper limit with $m\rightarrow2^m$. ...
0
votes
1answer
37 views

Infinite Sum Defined by $\int \frac{e^x}{x}dx$ vs. Exponential Function Taylor Series

Recently, when fiddling around with integration by parts, I noticed that it is possible to define infinite series that led to an integral. My calculus teacher noticed this, and told me to find $$ ...
1
vote
1answer
14 views

$a_n\geq b_n$ for $n>\bar{n}$ implies $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$

Consider two sequences of real numbers $\{a_n\}_n, \{b_n\}_n$. I know that if $a_n\geq b_n$ $\forall n$ then $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$. Suppose ...
2
votes
3answers
214 views

Showing that this sequence is eventually decreasing

I'm trying to show that this sequence $$a_n = \frac{3^n-7}{4^n+5}$$ is decreasing for all $n$ greater than some $N\in \Bbb N$. All I can see to do is something like $$a_{n+1} = ...
0
votes
1answer
32 views

Find the range of $x$ for which the sequence $\dfrac{n!} {k!(n-k)!}x^n $ converges to $0$ for a stabilised $k\in\mathbb{N}$

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $c$ and graded for ...
4
votes
3answers
96 views

Limit of $a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$

Find a limit of sequence: $$a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$$ $$a_1=0,a_2=0$$ I tried to prove that $a_n$ is bounded and monotonic, but I couldn't prove that $a_n$ is monotonic (by ...
0
votes
1answer
26 views

Is there any identity for this series?

While solving inequality and finite series problem I often come across this series- $$(n+1)(n+2)(n+3)...(n+n)$$. Is there a general solution to this form of a series? Thanks for any help!!
0
votes
0answers
16 views

Can you define a Cauchy product from this identity?

I can write vague expessions for $\zeta(3)$, the Apéry's constant, for example when I multipliy by $\frac{1}{n^4}$ the recursion relation (2) in page 2 here, and after I take the sum ...
1
vote
0answers
28 views

Closed form of an integral $\int_0^{\pi/2} \ln^n (\sin x) \, dx$

Let $n \in \mathbb{N}$. May we have a closed form for the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$ One obvious approach would be to go through beta functions and ...
2
votes
2answers
33 views

Can you get a closed-form for $\sum_{j=0}^{\infty}\frac{2^{2j-1}B_{2j}}{(2j)!}$?

Let $B_{k}$ the kth Bernoulli number, then using their asymptotic I can justify the absolute convergence of this series $$\sum_{j=0}^{\infty}\frac{2^{2j-1}B_{2j}}{(2j)!},$$ since, if there are no ...
1
vote
3answers
29 views

What is the general term of the sequence $u_{n+1}=c u_n+d$? [on hold]

What is the general term of the sequence $u_{n+1}=c u_n+d$ ?
1
vote
1answer
21 views

Error with the proof that all solutions to the Cauchy Functional Equation are linear

If $f(x)$ is continuous, it is known that $f(x+y)=f(x)+f(y)$ implies that $f(x)$ is linear, and non-continuous solutions are discussed in these links. (1, 2,3, 4) However, what is wrong with this ...
2
votes
2answers
59 views

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero?

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero? I was talking to a friend about it but neither of us were able to come up with anything. We're not sure if $\pi$ is essential ...
0
votes
3answers
33 views

Arc length of a sequence of semicircles.

Let $\gamma_1$ be the semicircle above the x axis joining $-1$ and $1$. Now divide this interval into $[-1,0]$ and $[0,1]$, and trace a semicircle joining $-1,0$ above the $x$-axis and another one ...
0
votes
2answers
22 views

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,…$

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,...$ I've tried to isolate n but its impossible to do.
0
votes
1answer
29 views

Bounded away sequence implications

Consider the sequence $\{\sqrt{n}|a_n-a|\}_n$ where $a_n, a \in \mathbb{R}$. Assume $\{\sqrt{n}|a_n-a|\}_n$ is bounded away from $0$ and $\infty$. Is this equivalent to or sufficient or necessary for ...
1
vote
1answer
14 views

Implications of $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$

Consider two sequences of real numbers $\{a_n\}_n$, $\{b_n\}_n$. Suppose $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$ where $\alpha \in \mathbb{R}$, $na_n\geq 0$ and big $O$ notation is explained ...
-2
votes
1answer
33 views

How to get from left to right-hand side of the equation? $ \sum_{k=0}^{d} \binom{2d+1}{k} = \frac{1}{2} \cdot 2^{2d+1} $

I would like to know how the left hand side of the equation is achieved. In particular why the $\frac{1}{2}$ is there. I don't understand how one can get from the left to the right side. $$ ...
1
vote
2answers
28 views

Cluster points and the sequence 1,1,2,1,2,3,1,2,3,4,1,…

I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a ...
0
votes
1answer
10 views

Geometric progression with reverse order

I have the following problem: Find three positive numbers which have the sum of $70$ and create a Geometric progression ($q>0$, increasing). Their inverse sum equals to $4/70$. Thank you!
2
votes
1answer
66 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
4
votes
2answers
39 views

Sequence bounded away from $0$ and $2$

Suppose I have a sequence of real numbers $\{a_n\}_n$ and I'm told that $\{a_n\}_n$ is bounded away from $0$ and $2$. (1) What does it mean exactly? My thinking is that it means $a_n\neq 0$ and $a_n ...
1
vote
1answer
37 views

Proving non-repitition of a sequence

I have heard that the sequence $$x_{n+1}=rx_n(1-x_n)$$ for $r$ between $3$ and $4$ does not recur i.e. there is no $a>0$ such that $x_{n+a} = x_n$ and $x_0$ is any number between 0 and 1 exclusive. ...
0
votes
0answers
39 views

all but one sub-strings within a cyclic string

over $GF(q)$ where $q\in\mathbb{N}$, we build a string of size $q^n-1$. now, how can I show that it is impossible to construct that string so it contains all sub-strings of size $n$ exactly once, but ...
1
vote
0answers
46 views

Estimate from above the series: $\sum_{i=1}^n \frac{1}{(i-j)^2}$.

Estimate from above the series: $$\sum_{i=1}^n \frac{1}{(i-j)^2}$$ for $n\in\mathbb N$, $j=1,2,\dots,n$ and $j\neq i$. I'd like to know an estimate dependent by $j$. This question comes from a simple ...
1
vote
0answers
19 views

Denominators of harmonic numbers: asymptotic behaviour.

About the sequence $d_n$ of the denominators of harmonic numbers, I know these facts: It is unbounded, since $p\mid d_p$ for any prime $p$. It contains only one $1$. What more is known? Specially, ...
0
votes
1answer
33 views

Given series diverges or converges?

Given: $$\sum_{i=1}^\infty \frac2{7*i + 21} $$ The limit of the $nth$ term is 0, it means we aren't sure if it diverges. On wolfram it says it diverges by comparison test, but how?
0
votes
1answer
19 views

Calculus Series and simplifying the expression

Considering the following series, Series How do I find a simplified expression for the ratio (an+1 / an)
0
votes
1answer
48 views

Maximum value of $\lambda$

It is given that a,b,c are be of same sign and a,b,c are in Harmonic progression i.e. $\frac{2}{b}=\frac{1}{a}+\frac{1}{c}$ and also $\frac{a+b}{2a-b}+\frac{c+b}{2c-b}\geq \sqrt{\lambda \sqrt{\lambda ...
3
votes
1answer
53 views

Prove if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges. No, $a_n, b_n$ are not necessarily ...
0
votes
3answers
20 views

Prove the existence of limit of certain sequences.

Problem: Let $0<a_1<b_1$ and $$a_{n+1}=\sqrt{a_n\cdot b_n},b_{n+1}=\frac{a_n+b_n}{2}.$$ Prove that $\{a_n\}$ and $\{b_n\}$ converge to some limit. Attempt: By induction and AM-GM, I can show ...
0
votes
0answers
11 views

Infinite series, continued fractions, nested radicals and such - is there a general recursive algorithm theory?

The nature of infinity and irrational numbers (among other things) always gave me trouble. But recently I learned to think about infinite sequences in terms of recursive algorithms and their time ...
-2
votes
2answers
63 views

Cant find the pattern [on hold]

These are two sequences: I tried looking for a ratio or difference but it wasnt working $$ 12,20,32,\underline{\quad},22,\underline{\quad},50 $$ $$ 6,10,16,8,\underline{\quad},22\underline{\quad} $$ ...
0
votes
0answers
35 views

Show that $\lim_{n\rightarrow \infty} (1-\omega(\frac{1}{n}))^n=0$ and $\lim_{n\rightarrow \infty} (1-o(\frac{1}{n}))^n=1$

Could you help me to show that (1) $\lim_{n\rightarrow \infty} (1-\omega(\frac{1}{n}))^n=0$ (2) $\lim_{n\rightarrow \infty} (1-o(\frac{1}{n}))^n=1$ where $o(\cdot)$ is little $o$ notation described ...
-2
votes
0answers
41 views

What is the sum of the series: $1+2(1/3)+3(1/3)^2+4(1/3)^3+\cdots$ [duplicate]

Please help me find the sum of this series. I tried separating the geometric series pattern from it but the remaining part is just the same.
0
votes
1answer
22 views

Limit of a function defined as the sum of a series

Given a decreasing sequence $a_n$ of positive real numbers, for $x>0$ define $$ f(x)=\sum_n \min\left(\frac{x}{a_n}, \frac{a_n}{x}\right). $$ Can $a_n$ be chosen so that $f(x)\to 0$ as $x\to 0$?