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

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2
votes
2answers
73 views

Prove $\sum_{n=1}^{\infty}\frac{n(3n−2)}{n!}=4e$

How to prove the following ? $$\sum_{n=1}^{\infty}\frac{n(3n−2)}{n!}=4e$$ I do know the series expansion for $e^x$.You may use it...
-7
votes
0answers
21 views

The sequence of nubers a1,a2,…an is defined as [closed]

The sequence of nubers a1,a2,...an is defined as an=(1/n+1)-(1/n=2) for each integer n>=1.what is the sum of he first 15 term of this sequence? a. 1/272 b. 1/6 c. 7/16 d. 1/2 e. 15/34 And explain ...
1
vote
2answers
27 views

General Term of a given series , Where $\sum^{n}_{r=1}U_r=\frac{3n}{2n+1}$

General term $U_r $ of a given series , Where $$\sum^{n}_{r=1}U_r=\frac{3n}{2n+1}$$ I can evaluate that $$ U_1=\frac{3}{3}$$ $$ U_2=\frac{1}{5}$$ $$ U_3=\frac{3}{35}$$ $$ U_4=\frac{1}{21}$$ $$ ...
3
votes
2answers
79 views

Problem concerning the sequence $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$

The question is : Prove that the sequence $\{s_n\}$ where $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$ is convergent. Hence find $\lim_{n \to \infty} \left(1 - 1/2 + 1/3 - ... - 1/2n\right)$. I have ...
2
votes
1answer
58 views

Sum of series $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$

I need to find sum of the series involving cube roots of unity $1−ω^2+ω^4−ω^6+ω^8−ω^{10}+ω^{12}+⋯+ω^{600}−ω^{602}+ω^{604}$. Found it in an old test paper. I applied Geometric Progression Sum Formula....
2
votes
1answer
49 views

Radius and Interval of Convergence $\sum_{n=0}^{\infty}\frac{7^n}{n!}x^n$

$\displaystyle \sum_{n=0}^{\infty}\frac{7^n}{n!}x^n$ I'm still trying to get the hang of these and feel like I've done something wrong here. After applying the ratio test I end up with: $\left|7x\...
3
votes
3answers
50 views

Radius and Interval of Convergence for $\sum_{n=1}^{\infty}\frac{5^n}{n^2}x^n$

$$\sum_{n=1}^\infty \frac{5^n}{n^2}x^n$$ After doing the ratio test I end up with: $$5|x| < 1$$ I'm confused, though, as to what is considered my interval of convergence and what is my radius. ...
3
votes
0answers
32 views

Question on possible Bilinearity of the Action of a Linear Functional on a Polynomial

I have been self studying the classical Umbral Calculus and have been reading works and papers from Rota and Roman on the material and I have a question regarding the following. The text uses $$\...
7
votes
1answer
38 views

The size of sets of positive integers not having distinct subsets with equal size and sum

Let us call a set $S$ of positive integers "good" if there does not exist a pair of distinct subsets $A,B\subseteq S$ who have equal size and an equal sum. Equivalently, a set $S$ is good if the ...
-2
votes
0answers
32 views

convergence of power series, expanded by maclaurin

I'm getting ready for a test and I stumbled upon a question which goes like this: A function f(x) is given and I need to expand it to a power series using Mcloren sequences and then calculate its area ...
0
votes
0answers
17 views

Convergence for $\sum_{\text{m composite}}\frac{e^{2\pi\sigma(m)i/(m+1)}}{m^s}$, where $\sigma(n)$ is the sum of divisors function

Let $\sigma(n)=\sum_{d\mid n}d$ the sum of divisors function. When one writes informally the identity $$\sum_{n=1}^\infty \frac{e^{\frac{2\pi\sigma(n)i}{n+1}}}{n^s}=1+\mathcal{P}(s)+\sum_{\text{m ...
0
votes
2answers
39 views

Finding rule of series

The answer is in terms of k. I tried finding a pattern between the consecutive terms of the series but could find none. Also I feel like there might be a systematic way for solving this that I do not ...
-2
votes
2answers
57 views

How many 4 digit numbers end in a 0? [closed]

I need to crack a code but all I know about it is that it has four digits and ends in zero.
5
votes
1answer
35 views

Given a finite sequence, can we always find a relation that generates that sequence?

This is just something I've been wondering about, but I have no idea what the answer is. I suspect it's yes. Given an arbitrary finite sequence, can we always find a relation that generates that ...
0
votes
0answers
36 views

D. F. Wallace's “Everything and more” $\S$7b : Cantor transfinite derivation from $P^{(n)}$

On $\S$7b of David Foster Wallace's book "Everything and more", the author explains how Cantor derived the concept of transfinite numbers from P, a second-species infinite point-set. "$P'$, can be "...
1
vote
1answer
18 views

Interchange of limit and sum justification

From the book I have been reading, it seems the following result is implicitly used: If $f_{a}(x):=\sum_{n=0}^{\infty} s_{n}(a,x)$ converges uniformly with respect to (large) $a$, for example $\...
0
votes
2answers
17 views

Alternating series test and divergence test similair?

The alternating series test requires: - Bn to be decreasing - lim Bn (to infinity) to be 0 In my book I see examples where the series fit the first one, but does not fit the second one and then they ...
1
vote
1answer
40 views

Solving series involving Poisson and Binomial

I'm trying to solve $$ \sum_{b=0}^\infty Poisson(b, \lambda)\sum_{x=0}^b binomial(x, b, p)\left(\frac{x}{x+1}\right)^x \\ = \sum_{b=0}^\infty \frac{e^{-\lambda} \lambda^b}{b!}\sum_{x=0}^b {b \choose ...
1
vote
1answer
27 views

Comparison test for alternating series

Give $\sum_n (-1)^n a_n$ where $a_n \geq 0$, if we have that $$0\leq c_n \leq a_n \leq b_n,$$ $c_n, b_n$ are monotone decreasing and go to zero, thus $\sum_n (-1)^n c_n$ and $\sum_n (-1)^n b_n$ both ...
0
votes
0answers
16 views

set-theoretic definition of sum of sequences, difference of sequences, product of sequences,…

I have $f,g \in \Bbb R^\Bbb N$, I define: $(f+g):=\{(y,z)| z= f(y)+g(y) \}$ $(-f):=\{(y,z)| z=-f(y) \}$ $(f-g):=f+(-g)$ $(f\cdot g):=\{(y,z)| z= f(y)\cdot g(y) \}$ Is it correct? I have problem ...
2
votes
2answers
33 views

Is the partial sum of cosine bounded?

Is it true that $\sum_{k=1}^n \cos k$ is a bounded sequence? If so how to prove? I want to prove the series of $\cos n/(\sqrt n)$ is convergent by abel test but I dont know if the partial sums of ...
3
votes
1answer
51 views

How to determine sum of an alternating power series and to prove that sum is positive

I am working on a problem involving an alternating power series as follows: $$\sum_{i=0}^{a-2} (-1)^{a+b-i-2}(a+b-i-1)x^{a+b-i-2}$$ $a$ and $b$ is constant with $0<x<1$ I would like to ...
22
votes
2answers
259 views

Fractal dimension of the function $f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}$

Consider the function $$ f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}\, . $$ This is a bizarre and fascinating function. A few properties of this function that SEEM to be true: ...
1
vote
2answers
14 views

Dealing with a Sequence of Sets with Two Indices and Simple Function based on that Sequence of Sets

Okay so I have a measurable function $f$ and a set $E_{n,i}$ $$E_{n,i}=\left\{ x:\frac{i-1}{2^{n}} \leq f(x)<\frac{i}{2^{n}}\right\}$$ where $i=1,...,n2^{n}$ and $n=1,2,...$ Then I have another ...
0
votes
0answers
28 views

On Dirichlet series and Firoozbakht's conjecture

On assumption of the Firoozbakht's conjecture (this is the Wikipedia, but the reference is for Carlos Rivera's Page) one has that can writes informally the Dirichlet series in LHS of this inequality $$...
5
votes
2answers
108 views

A closed form of the series $ \sum_{n=1}^{\infty} q^n \sin(n\alpha) $

I am having problems with the following series: $$ \sum_{n=1}^{\infty} q^n \sin(n\alpha), \quad|q| < 1. $$ No restrictions on $\alpha$. I need to find out whether it converges and if yes, ...
1
vote
0answers
15 views

What's about $\sum_{n=1}^\infty\frac{\mu(n)}{n}f(X^{\frac{1}{n}})$, where $\mu(n)$ is the Möbius function?

Let $X=\sigma+it$ the complex variable, and $\mu(n)$ the Möbius function. Inspired in Riemann function $R(X)$ I would like to ask you Question. What conditions are required to be satisfied by a ...
0
votes
1answer
19 views

Construct a function pertaining to the OEIS sequence A131229 (Numbers congruent to {1,7} mod 10)

OEIS sequence A131229 ("Numbers congruent to {1,7} mod 10") begins $\{1, 7, 11, 17, 21, 27, 31, 37, 41, 47, 51,...\}$. I want a function $f(x)$, specifically such that $f(\frac{1}{2}) =\frac{7}{2}$, ...
1
vote
1answer
47 views

Show $f(x) = \sum_{n=1}^\infty \frac{\sin{nx}}{n^{5/2}}$ is $C^1$

The task is to show $$f(x) = \sum_{n=1}^\infty \frac{\sin{nx}}{n^{5/2}}$$ is a continuous function on $\mathbb{R}$ with a continuous derivative. Showing that the series converges at each point is ...
1
vote
1answer
39 views

Condition on $(a_n)_{n \in \mathbf{N}}$ for Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ to converge for $Re(s) > 1$.

I'm curious about the conditions on a sequence $(a_n)_{n \in \mathbf{N}}$ of real numbers such that the Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ converges absolutely for $Re(s) > 1$. ...
1
vote
0answers
21 views

simplification of functional sum

I have the following infinite sum which seems to be familiar so I was wondering if it admits a possible simplification ! $$ \sum_{k\geq 0}\left( g(x_k,x_{k+1})-g(x_k,x_k)\right) $$ where $g$ is only ...
1
vote
0answers
29 views

Quotient of Confluent Hypergeometric Functions of the 1st Kind

I want to solve the following problem for x: \begin{equation} \frac{\mathrm{d}}{\mathrm{d}x}\ e^{-\beta_{1}x}\,{_{1}}F_{1}[-\alpha_{1};-\alpha_{3};\beta_{3}x]=0 \end{equation} where, $\alpha_{1},\...
1
vote
3answers
57 views

nature of the series $\sum \tfrac{(-1)^{n}\ln(n)}{\sqrt{n+2}}$

I would like to prove the following series convergent $$\dfrac{(-1)^{n}\ln(n)}{\sqrt{n+2}},\quad \dfrac{(-1)^{n}\ln(2)}{\sqrt{n+2}}$$ using Alternating series test: $u_n=\dfrac{(-1)^{n}\ln(n)}{\...
2
votes
0answers
41 views

Multiplication of polynomials of the same degree

Consider polynomials of the form \begin{equation} p(x)=x^{n-2r}\sum_{i=0}^ra_ix^{2i}, \end{equation} where \begin{align} r&=n/2, \quad n \quad \text{even},\\ r&=(n-1)/2, \quad n \quad \text{...
-1
votes
2answers
69 views

Prove that a sum of continuous functions is continuous

$\forall n\in N$, let $f_n(x): [0,1]\rightarrow \mathbb{R}$ be continuous functions and $M$ a positive integer. If $\forall x \in [0,1]$, $\lvert\sum_{n=1}^{\infty} f_n(x)\rvert \lt M$, then $\sum_{...
2
votes
1answer
48 views

Is this infinite series of continuous functions $f(x)=\sum_{n=1}^{\infty} \sin(\frac{x}{n^2})$ continuous?

The original question: Consider the function $$f(x)=\sum_{n=1}^{\infty} \sin\left(\frac{x}{n^2}\right).$$ Is $f$ a continuous function on $\mathbb{R}$ ? I know that the infinite sum of continuous ...
0
votes
2answers
80 views

About bound of series

$$ Can anyone please tell me bound for following series I simplified this upto $$(2n-1)-2\left( \frac{1}{n+1}+\frac{2}{n+2}+\ldots+\frac{n-1}{2n-1} \right)$$ Also I get one of my bound as $\frac{n(...
4
votes
2answers
91 views

The convergence of $\sqrt{x^2+\frac{1}{n}}$

Let $f_n(x)=\sqrt{x^2+\frac{1}{n}}$. i) Determine the limit-function $f$. ii) Does $f_n(x)$ converge uniformly to $f$? For the first: We have $\lim_{n\rightarrow \infty}\sqrt{x^2+\frac{1}{n}}=\sqrt{...
3
votes
0answers
205 views
+50

Explicit solutions to a digamma function equation

My main question: Can we obtain the exact solutions from the following equation? $$ \sum_{k=1}^{n}\cfrac{1}{k-x-1}=0 $$ Notation: This problem was reached from the digamma function $\psi$ as ...
0
votes
0answers
26 views

Deriving the sequence if the generating function is irreducible?

I am trying to better understand generating functions and how they can be derived / manipulated / etc. Right now I am operating on this identity, slightly modified from the answer here: For a ...
0
votes
2answers
56 views

Prove $\lim _{ n\rightarrow \infty }{ { x }_{ n }^{ k } } ={ \left( \lim _{ n\rightarrow \infty }{ { x }_{ n } } \right) }^{ k }$

I'm trying to prove that the limit of the sequence $x_n^k$ is the same as the limit of $x_n$ all raised to the $k$th power. Prove $$\lim _{ n\rightarrow \infty }{ { x }_{ n }^{ k } } ={ \left( \lim ...
1
vote
2answers
34 views

Summation of a convergent series

I have the following problem: So I start as follows: $B_{2}M_{2}=\frac{1}{\sqrt{3}}$ and I realize that $B_{2}M_{2} = A_{2}B_{2}$, so $B_{3}M_{3} = \frac{1}{\sqrt{3}}^{2}$. Next, I compute $A_{1}...
5
votes
1answer
114 views

How do we prove that $4(3\sqrt2-4)=\prod_{n=1}^{\infty}\left({e^{2\pi(2n-1)}-1\over e^{2\pi(2n-1)}+1}\right)^8?$

How do we prove that $$4(3\sqrt2-4)=\prod_{n=1}^{\infty}\left({e^{2\pi(2n-1)}-1\over e^{2\pi(2n-1)}+1}\right)^8\tag1$$ Rewrite as, to keep it simple Let $a=e^{2\pi(2n-1)}$ $$4(3\sqrt2-4)=\...
0
votes
0answers
44 views

Positive function(or series) that tends to 0 at infinity

Consider these two questions: -If we have an ultimately positive function that tends to 0 at infinity, does this imply that the function has to be decreasing? -Similar questions but for series: if ...
1
vote
0answers
35 views

Convergence of sequence $a_0 := b$ and $a_{n+1} = 2^{a_n}$ in $\hat{\mathbb{Z}}$.

I have a question about the convergence properties of a sequence in $\hat{\mathbb{Z}}$, the completion of $\mathbb{Z}$. It is part of an exercise is due to this syllabus. I got confused somewhere. ...
0
votes
0answers
16 views

On a second set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of these claims and show where were my mistakes or inaccurancies? Also ...
0
votes
0answers
14 views

On a first set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of this claims and show where were my mistakes or inaccurancies? Also ...
1
vote
1answer
27 views

Recurrence equation for sequence of vectors

Consider recurrent formula for a sequence of numbers $(y_n)$ (either real or complex): $$a_k y_{n+k}+a_{k-1}y_{n+k-1}+\cdots+a_0y_n=\sum_{i=0}^k a_i y_{n + i} = 0$$ It's known that the exact explicit ...
0
votes
1answer
57 views

Prove $\sum_{i=1}^{n}\frac{i}{(i+1)!}= 1-\frac1{(n+1)!}$ [duplicate]

Required to prove: $1-\frac1{(k+2)!}$ $$\begin{align*} \sum_{i=1}^{k+1}\frac{i}{(i+1)!}&=\sum_{i=1}^k\frac{i}{(i+1)!}+\frac{k+1}{(k+2)!}\\ &= 1-\frac1{(k+1)!}+\frac{k+1}{(k+2)!}\tag{induction ...
2
votes
3answers
57 views

Convergence test of $\sum\limits_{n=3}^{\infty} \frac{1}{n\log(n)\log(\log(n))}$

I know, there are some threads dealing with this sum but I want to solve it with the integral test for convergence(more) $$\sum\limits_{n=3}^{\infty} \frac{1}{n\log(n)\log(\log(n))}$$ I can't ...