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

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11
votes
0answers
100 views

Sum the series $\sum_{n = 0}^{\infty} (-1)^{n}\{(2n + 1)^{7}\cosh((2n + 1)\pi\sqrt{3}/2)\}^{-1}$

In one of his letters to G. H. Hardy, Ramanujan gave the following sum $$\dfrac{1}{1^{7}\cosh\left(\dfrac{\pi\sqrt{3}}{2}\right)} - \dfrac{1}{3^{7}\cosh\left(\dfrac{3\pi\sqrt{3}}{2}\right)} + ...
1
vote
0answers
47 views

Series of polynomials and uniformly convergence

It's part of the proof of a Lemma of an article I was reading (Algebraic values of transcendental functions at algebraic points). I couldn't understanding one thing: Let f be a complex function such ...
3
votes
2answers
146 views

Prove that if an infinite series converges, then the associative property holds

I'm self-studying from the book Understanding Analysis by Stephen Abbott and have no idea how to do exercise 2.5.2 on page 57. The exercise is as follows: Prove that if an infinite series converges, ...
2
votes
1answer
47 views

Is $\sum_{n=1}^N e^{2 \pi p_n z i}$ bounded for irrational $z$?

Let $p_n$ be the $n$th prime number. If $z$ is irrational real, is it known whether the partial sums $\sum_{n=1}^N e^{2 \pi p_n z i}$ are bounded? It seems the partial sums are unbounded if $z$ is ...
4
votes
1answer
85 views

Compute the following series $\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}$

Does the following series have a 'closed' form : $$\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}.$$ Where $n\in \Bbb{N}$ and $a,b \in (0,+\infty)$ For $a,b$ integer we can use Partial fraction ...
3
votes
2answers
96 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
0
votes
1answer
41 views

Does every convergent sequence have a sub-sequence whose terms comes closer than any positive sequence?

Let $(x_n)$ be convergent sequence of real numbers and $(y_n)$ be any sequence of positive real numbers , then is it true that there is a sub-sequence $(x_{r_n})$ such that ...
2
votes
3answers
81 views

Landau's proof that $\zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}$

In the Handbuch, Landau proves that for all $s>1$ the following equality holds $$\zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}.$$ I'm having trouble with the following part of his proof: Landau says that ...
3
votes
0answers
49 views

Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
4
votes
1answer
44 views

Is Banach space a correct context to study sequences and series?

Recently, I have reviewed elementary analysis and I realized that every theorem in the text(Rudin-PMA) about series can be generalized to Banach space. Here is an example. Below is the theorem ...
5
votes
3answers
710 views

Determining if a recursively defined sequence converges and finding its limit

Define $\lbrace x_n \rbrace$ by $$x_1=1, x_2=3; x_{n+2}=\frac{x_{n+1}+2x_{n}}{3} \text{if} \, n\ge 1$$ The instructions are to determine if this sequence exists and to find the limit if it exists. I ...
0
votes
1answer
41 views

Convergence of $\sum_{n=1}^{\infty} n$ and integral test [duplicate]

I have read online, that one can show that $\sum_{n=1}^{\infty}n = -\frac{1}{12}$. But isn't this a Riemann Series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p=-1$. And if so, can't ...
0
votes
1answer
74 views

Does it converges? $ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $

As I checked on Wolfram Alpha I know that $$ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
2
votes
3answers
94 views

Finding the second derivative of an infinite series

I'm asked to find the 2nd derivative of $$f(x)=-2x+\frac{2x^3}{3!}-\frac{2x^5}{5!}+\cdots+(-1)^{n+1}\frac{2x^{n+1}}{(2n+1)!}+\cdots=\sum \limits_{n=0}^{\infty} \frac{(-1)^{n+1}2x^{2n+1} }{(2n+1)!}$$ ...
2
votes
1answer
66 views

Complex Analysis Weierstrass M-Test

Prove that each of the following series converges uniformly on the corresponding subset of $\mathbb C$: $$\begin{align*} \text{(a)} \; & \sum_{n=1}^\infty \frac{1}{n^2 z^{2n}}, & & ...
1
vote
4answers
131 views

Finding the sum of a series till $n$ terms

Series: 5, 11, 19, 29, 41 Find the sum of the series up to $n$ terms. Well the method that comes to my mind is to find the nth term of the sequence, and then find their summation. I use the basic ...
2
votes
0answers
51 views

$f\in C^\omega ((a-R,a+R),\mathbb{R})$ [closed]

We discuss the following question in the field of real numbers . A a power series $f(x)= \sum_{n=0}^\infty a_n (x-a )^n$ converges in $(a-R,a+R).$ Prove: $$\forall x_0\in\left(a-R,a+R \right), ...
1
vote
1answer
28 views

Recovering a continuous function from a discrete one.

Consider a well-behaved function $f(x)$ defined on $x\geq0$, and construct a discretized version of it using the Dirac-delta function: $$ ...
0
votes
1answer
45 views

Rearrangement Theorem Applications [closed]

1) Suppose that $\sum a_n$ converges and $a_n > 0$ for all $n > 1$. Prove that $\sum a_n^3$ also converges. 2) Show an example of a series $\sum a_n$ that converges, while $\sum a_n^3 ...
0
votes
1answer
51 views

$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$

$$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$$ I found this question during my study. In my opinion ,it is not difficult to solve ,but it is interesting. So I ...
1
vote
3answers
58 views

How to calculate the integral of $\sum_{n=1}^\infty (1/r)^{n+1} r^2$?

How to calculate this integral? $$\int\limits_0^1 {\sum\limits_{n = 1}^\infty {\left( {\frac{1}{r}} \right)} } ^{n + 1} r^2 dx$$ Here $r$ is a real number
0
votes
1answer
34 views

Power of matrix expansion

We know about the expansion $(a+b)^n\tag 1$,for scalar variables. What will be the equivalent when we want to find $(A+B)^n \tag 2 $, when A and B are square matrices? Can we treat it as same as ...
10
votes
3answers
404 views

Consecutive Prime Gap Sum (Amateur)

List of the first fifty prime gaps: 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4. My ...
4
votes
1answer
94 views

'Deriving' the Laplace Transform from the $z$ Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
0
votes
1answer
55 views

Calculation of a series

Calculate the series $$\sum^{\infty}_{n=1}\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}$$ I tried the root test but couldn't figure out, any ideas?
4
votes
5answers
178 views

Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent?

Is this series convergent or absolutely convergent? $$\sum _{n=1}^{\infty }\:(-1)^n \frac {1} {n^2}$$ Attempt: I got this using Ratio Test: $$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$
0
votes
1answer
36 views

$\Delta^kn^\alpha$ converges monotonically to zero when $\alpha<k$

I am trying to prove that the $k$-th finite difference of the series $n^\alpha$ converges to zero monotonically as $n\to\infty$ when $\alpha<k$. The differential analogue is ...
0
votes
3answers
47 views

Question on Sequences and limits

If sequence {$a_n$} satisfies $\displaystyle \lim_{n \to \infty} (2n-1)a_n=40$, what is the value of $\displaystyle \lim_{n \to \infty}na_n$ ? Any hints ?
1
vote
5answers
66 views

Inequality involving a finite sum

this is my first post here so pardon me if I make any mistakes. I am required to prove the following, through mathematical induction or otherwise: $$\frac{1}{\sqrt1} + \frac{1}{\sqrt2} + ...
1
vote
4answers
89 views

Can this limit be solved in this manner?

$\lim _{n\to \infty }\left(\left(1+\frac{1}{1}\right)\cdot \left(1+\frac{1}{2}\right)^2\cdot ...\cdot \left(1+\frac{1}{n}\right)^n\right)^{\frac{1}{n}}$ Hi. I'm trying to solve this limit without ...
2
votes
1answer
74 views

General term of sequence $a_0=2$ and $a_{n+1}=2a_n^2-1$

Is it possible to find general term of this sequence? $a_0=2$ and $a_{n+1}=2a_n^2-1$
1
vote
1answer
189 views

How to find a series for comparison with $\sum 1/\sqrt{n(n+1)}$?

The series $$\sum_{n=1}^{+\infty} \frac{1}{\sqrt{n(n+1)}}$$ I have tried ratio and integral both lead me to inconclusive, so probably it's by comparisson but I can't find What to compare.
2
votes
4answers
57 views

Alternative way to solve this limit?

The solution to this limit should be 3. I know that it can be solved by using the squeeze theorem, by coming up with two other sequences whose limit is 3, but I would prefer some other method if ...
20
votes
1answer
265 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
0
votes
0answers
26 views

analyticical solution of a Recurrence serie

I have the following recurrence relation: $L_N = L_x + \frac{1}{\frac{1}{L_r} + \frac{1}{L_{N-1}} }$ $L_0 = L_x +L_r$ If $L_x$ and $L_r$ are kept constant, I have no problem to find a solution to ...
1
vote
5answers
104 views

Existence of limit for a given sequence: $x_{n+1} \le x_n + 1/{n^2}$

Let $x_n$ be a sequence in $\mathbb{R}$ such that $$x_{n+1} \le x_n + \frac{1}{n^2}$$ Prove that $\lim x_n$ exists (it can be a real number or infinite). I've tried to prove it using the ...
1
vote
2answers
51 views

Summation of $(((2N+1).2 + 1).2 + 1)\cdots $

Is there a way to sum up this series: $(((2N+1).2 + 1).2 + 1)\cdots $ The actual question that I encountered was on a coding site (HackerRank) where it said that you had a tree which grows twice in ...
0
votes
1answer
21 views

Approximate average customer lifetime using churn

Average customer lifetime can be approximated using formula: $1/Churn$ (churn is a real number in interval $(0;1)$ The same can be achieved using more complex formula: $\sum_{n=0}^\infty ...
0
votes
2answers
105 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
0
votes
1answer
40 views

Sequence Covergent $\iff$ absolutely convergent

Is it correct that the sequence $a_n \rightarrow 0 $$\iff |a_n|\rightarrow 0$? I have it written down but I am unaware where I got it from. If so is it true for any limit?
3
votes
2answers
75 views

$\lim \sqrt[n]{a^n + b^n}$

I've seen some answers here for why this limit is the maximum between $a$ and $b$, but all of then included the hypotesis that $a$ and $b$ are both non negative. It was asked to show that this limix ...
1
vote
1answer
27 views

Build a sequence, $a_n$ with $PL=\{0,1,2\}$

I was asked to build a sequence which has exactly three partial limits: $\{0,1,2\}$. Also, for every $n\in\Bbb{N}: \left|a_{n+1} - a_n\right| < 1$ At first I thought about: $$a_n = \begin{cases} ...
1
vote
1answer
59 views

Is the calculation of the series in this video correct?

I am watching this video (from MIT OCW) and Prof. Jerison is explaining about series. He is trying to calculate that if some blocks of equal length are kept on top of each other, will the last block ...
2
votes
2answers
67 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
3
votes
2answers
174 views

Solve the “two trains and a fly” problem the hard way

Few days ago, I was asked the following question: There are $2$ cities. city $A$ and city $B$ with distance $d$=600km There are $2$ trains with speed of $vt$ = 100km/h. There is $1$ fly with speed of ...
2
votes
2answers
78 views

How should I go about solving the following limit?

How do I solve the following limit? $$\lim _{x\to \:1} \frac{\left(1-x^{1/2}\right)\left(1-x^{1/3}\right)\cdots \left(1-x^{1/n}\right)}{(1-x)^{n-1}}$$ The solution should be 0, but the source isn't ...
1
vote
1answer
35 views

For what p is the series abolsutely convergent and conditionally convergent?

My lecturer has a passion for logs, and I'm reviewing some of her past papers and I found this question and I'm having a quite a difficult time dealing with, any help would be appreciated $$ ...
1
vote
2answers
48 views

Taylor series convergence for sin x

a. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}<\sin x<x-\frac{x^3}{3!}+\frac{x^5}{5!},$ b. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots-\frac{x^{4k-1}}{(4k-1)!}<\sin ...
2
votes
1answer
90 views

Explanation of the formulas for sums $\sum nr^n$ and $\sum n^2 r^n$

So I am studying series for an exam right now and there is an example in the book I am studying (unfortunately the book is specific to my university so I cannot give any link) where certain series' ...
3
votes
2answers
116 views

Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$ I tried ratio test but it became complicated, I have never seen such radius of convergence problem ...