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

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3
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0answers
21 views

FRACTRAN for natural numbers

Is there a simple analogue of FRACTRAN that maps a natural number to a natural number, instead of mapping a list of fractions to a natural number? One could use Gödel encoding to translate FRACTRAN ...
1
vote
1answer
30 views

What is the definition of the absolute convergence of an infinite product $(1+a_1)(1+a_2)(1+a_3)\cdots$?

For an infinite product $(1+a_1)(1+a_2)(1+a_3)\cdots$, whats the definition of convergence and absolute convergence? Why the absolute convergence corresponding to the absolute convergence of sum ...
0
votes
1answer
18 views

What is the difference of the greatest of the limits $\overline{\lim}_{n\to \infty}$ and the least of the limits $\underline{\lim}_{n\to \infty}$??

What are exactly these three limits for an infinite series $x_n$? $$\overline{\lim_{n\to \infty}} x_n$$ $$\underline{\lim}_{n\to \infty} x_n$$ $$\lim_{n\to \infty} x_n$$ Can they be different from ...
0
votes
0answers
20 views

Calculat sums of the form $\lim_{n\rightarrow\infty}\sum_{i=0}^{n}\left(\dfrac{i}{n}\right)^{f(n)}$

Problem: calculate the sums of the form: $$\lim_{n\rightarrow\infty}\sum_{i=0}^{n}\left(\dfrac{i}{n}\right)^{f(n)}$$ Inspiration: one problem lets us prove that ...
0
votes
2answers
32 views

Trying to understand why 2 times the sum of consecutive integers from 0 to n is equal to n times n+1

I am sorry if this question ends up being a duplicate, as I am having a bit of a challenge explaining it to myself well enough to know how to query it. There is a Facebook meme that has been ...
0
votes
0answers
14 views

Intriguing Poisson sum with hyperbolic function

I've been playing with lots of Poisson sums lately, and I thought this one to be interesting:\ $$\sum_{k\in\mathbb{Z}}\left(\frac{1}{(k+x)\sinh{(k+x)\pi q}}-\frac{1}{\pi q (k+x)^2}\right)$$I want to ...
1
vote
6answers
60 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...
2
votes
1answer
29 views

Finding a general term for the sequence

Find a general term in simplest form for the sequence: 2, 1, -4, 7, -10, 13, -16 This is what I tried: $a_n = a_1 + (n - 1)*d$ $a_n = 2 + (n-1)*3$ $a_n = (-1)^n (2+(n-1)*3)$ which doesn't work. I am ...
0
votes
0answers
12 views

Double Sum of Series Unchanged When Each Term Scaled

Can the following ever hold: $$\sum_{i}\sum_{j}(-1)^ja_{i,j}=\sum_{i}i(b_{i})\sum_{j}(-1)^ja_{i,j}$$ where $b_{i}>1/i$ for all $i$? What if you tack on the fact that ...
3
votes
1answer
39 views

Sum involving the number of zeros of $k$

I'm currently interested in sums involving digit-functions. Especially, I'd like to calculate the following sum: $$ s=\sum_{k=1}^{\infty}{\frac{a_0(k)}{k\left(k+1\right)}} $$ Where $a_0(k)$ is the ...
0
votes
2answers
24 views

Series And Sequences Question

Can someone help show me what I did wrong? The question is "Find the sum of the first ten terms in this geometric series: $-5, 10, -20, \ldots$ I plugged it into this equation: $S_n = a(r^n ...
0
votes
2answers
62 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
1
vote
0answers
23 views

Proving an inequality involving discrete variables

I'm trying to show that the following inequality holds $$ \frac{1-x^{n}}{1-x^{n+1}}\geq\frac{\sum_{i=0}^{n-2}x^{i}(1-x_{1}^{n-(i+1)})}{\sum_{i=0}^{n-1}x^{i}(1-x_{1}^{n-i})}, $$ where $n$ is a ...
2
votes
1answer
50 views

Notion of fixed point for the sequence $x_{n+1}=f_n(x_n)$

Let $(X,d)$ be a complete metric space and let $f_n:X\to X,\ n\in \mathbb{N}$ be a sequence of contractions. Is there any notion of fixed point for the following sequence? $$x_{n+1}=f_n(x_n),\ ...
1
vote
1answer
49 views

Rearranging a series' terms

So i am asked to rearranje the terms in this series: $$ \sum_{i=1}^\infty \frac{(-1)^{n+1}}{n} = 1- \frac 12 +\frac13-\frac14+... $$ so that the sum of the series is equal to 0. I've seen the ...
2
votes
1answer
46 views

Find a constant $C$ such that $ \Bigg| \frac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n$

Consider the following: $$ \Bigg| \dfrac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n $$ How to find an expression for $C$ independent of $k$ and thus $n$? It arises ...
2
votes
2answers
47 views

How to show that the following series converges to 1

Let $f$ be a function on $\mathbb{R}$, non-zero only on $[0,2)$. In particular $f(x)=1,x\in[0,1]$ and decreasing to zero, starting from $x=1$. Let $g(x)=f(x)-f(2x)$. Show that $$\sum_{j=0}^\infty ...
0
votes
2answers
67 views

The Laurent Series of $\dfrac{e^z}{z^2-1}$

The Laurent Series of $\dfrac{e^z}{z^2-1}$ At $z=1$ As we seek for powers of $z-1$, note that: $$e^z=e\cdot e^{z-1}=e(1+(z-1)+\dfrac{(z-1)^2}{2!}+\dfrac{(z-1)^3}{3!}+...)$$ So: ...
3
votes
1answer
42 views

Does this sequence of functions converge uniformly?

So the questions says, let $a_n$ be a sequences of real numbers such that $\limsup |a_n| = 0$. Let $X = [0, 1]$ and for each $n \in \mathbb{N}$ the function $\space$ $f_n :$ $X \mapsto ...
0
votes
0answers
12 views

What are the coefficients of Modified Fourier Bessel series?

What are the co-efficients of Modified Fourier Bessel series? $$\phi g(r,z,Φ) = \sum_{v=1}^∞ \sum_{m=1}^∞ \sin\left(\frac{mπz}{L}\right) I_v \left(\frac{mπr}{L}\right)\left(A_{vm} \cos(vΦ) + B_{vm} ...
3
votes
1answer
55 views

Proving that a sequence converges or diverges [on hold]

Prove or disprove that there is a sequence $n_k$ of positive integers (that is not constant) such that both $\cos(2n_k!)$ and $\sin(2n_k!)$ converge. I think that the series diverges but I am not ...
0
votes
1answer
132 views

An Infinite series I

By decompising fractions one can show that \begin{align} \sum_{n=1}^{\infty} \frac{1}{n \, (n+1)^{2} \, (n+3)} = \frac{65}{72} - \frac{\zeta(2)}{2}. \end{align} The fraction can also be seen in the ...
1
vote
1answer
37 views

$ |x_{n} - y_{n}| < \frac{1}{n} \Rightarrow |x'_{n} - y'_{n}| < \frac{1}{n}$

This came from a proof on uniform continuity theorem. My textbook claims that if sequence $(x_j)$ and $(y_j)$ on a compact set D have the condition that $ |x_n - y_n| < \frac{1}{n}$, and $(x'_j)$, ...
2
votes
3answers
104 views

Find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$

How to find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$ in the formal way? Numerically its value is $\approx 0.0217326$ and the partial sum formula contains the first derivative of ...
0
votes
1answer
17 views

Finding the bounds for a truncation error

I have two series, $S$ and $T$ which approximate $\pi$ such that $$S_n = 4 \sum_{i=1}^n \cfrac{-1^{i+1}}{2i-1}$$ and $$T_n = \Big(12 \sum_{i=1}^n \cfrac{-1^{1+i}}{k^2} \Big) ^{\frac{1}{2}}$$ It is ...
0
votes
0answers
12 views

What does “empirical error” mean in this context?

I recently sat an exam for computational mathematics. The question asked for us to: "Write the empirical error in $\mathcal{O}(n^{-p})$ where $p$ is some integer" We were given a series $$S = 4(1 - ...
1
vote
1answer
37 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \le 0$. Proof: I will attempt to show that the ...
6
votes
5answers
132 views

Show that $(1+p/n)^n$ is a Cauchy sequence for arbitrary $p$

It is a generalization of this question. I am looking for a similar derivation as in here. Can we prove that $(1+p/n)^n$ is a Cauchy sequence for any $p \in [a, b]$ by showing that $$ \Bigg| \left( ...
1
vote
2answers
33 views

How to show that this interesting difference of products is $O \left( \frac{1}{n^2} \right) $

Let $k \leq n$. Consider the following difference of products: $$ \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n+1} \right) - \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n} \right)$$ For $n=1,2,3$, this is ...
1
vote
2answers
55 views

What textbooks should I use for Trigonometry and Calculus? My basics are terrible.

I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by ...
1
vote
1answer
22 views

Summation of infinite series, where difference in consecutive denominator forms an A.P.

What is the sum of an infinite series where each term can be written as $\frac{p}{q}$, where p=1 always the difference between 2 consecutive denominators forms an A.P. for example $\frac{1}{2}$, ...
0
votes
2answers
12 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
2
votes
2answers
47 views

What's special about the cauchy product?

I am reading chapter 3 in Rudin's Principles of Mathematical Analysis. In it, Rudin defines the Cauchy product of two series. That is, given $\sum a_n$ and $\sum b_n$, $c_n=\sum_{k=0}^n a_k b_{n-k}$ ...
0
votes
2answers
37 views

Sum of Converging sequence

I'm given this sequence where it goes: $$ 1,\; \frac1a,\; \frac1{a(a+b)},\; \frac1{a^2(a+b)},\; \frac1{a^2(a+b)^2}, \frac1{a^3(a+b)^2}, \dotsc $$ where $a$ and $b$ are any positive integers How ...
1
vote
1answer
44 views

Stadium Seating - Geometric Sequences

A circular stadium consists of sections as illustrated, with aisles in between. The diagram show the tiers of concrete steps for the final section, Section K. Seats are to be place along every step, ...
1
vote
1answer
33 views

Asymptotic Estimates for a Strange Sequence

Let $a_0=1$. For each positive integer $i$, let $a_i=a_{i-1}+b_i$, where $b_i$ is the smallest element of the set $\{a_0,a_1,\ldots,a_{i-1}\}$ that is at least $i$. The sequence ...
2
votes
1answer
22 views

Absolute value of infinite series sum

How does it come about that $$\left|\Sigma_{n=-N}^{N}c_n(f)e^{inx} - \Sigma_{-\infty}^{+\infty} c_n(f)e^{inx}\right| = \left|\Sigma_{|n|>N} c_n(f)e^{inx}\right|?$$ What happens with the $n$-index? ...
1
vote
2answers
27 views

series comparison test

Is this correct? Q:Determine $\sum_1^n$$\frac{2}{3+5n}$ converges or diverges. A:$\frac{2}{6n}$ < $\frac{2}{3+5n}$ , since $\sum_1^n$$\frac{2}{6n}$ is a harmonic series and diverge, then ...
4
votes
2answers
99 views

Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$?

I hope this is not a duplicate question. If we modify the well known geometric series, with $a>1$, to $$ \sum_{n=0}^{\infty} \frac{1}{1+a^n} $$ is there a closed form with a name? I suspect ...
1
vote
0answers
26 views

Remainder Estimate for Integral test

I have the following question, it is a fill in the blank type question, however when I submit my answer, the system which verifies it say it is incorrect. I believe I am right, so I was hoping for ...
5
votes
0answers
113 views

Infinite Product Representation of $\sin x$

I've recently taken interest in infinite products, and I'm having trouble with a proof I found in this PDF file: "Infinite Products and Elementary Functions": An intermediate step in finding an ...
1
vote
1answer
67 views

Question about two sequences with a common limit

Suppose $a _n$ is a sequence of positive integers such that $ \lim_{n \rightarrow \infty} (a_n)^{\frac{1}{n}} $ exists. Suppose there exists a sequence of positive integers $ b_n $ such that $$ a_n = ...
0
votes
0answers
33 views

Additional explanation needed for the solution to a spesific sequence

Some of my attempts include trying to use the the term formula for geometric sequences and some other manipulations in hope of getting a more clearer, workable expression, though without success. ...
0
votes
3answers
83 views

Find the formula of the sum of $\frac{1}{1+x^2} + \frac{1}{(1+x^2)^2} + \dots + \frac{1}{(1+x^2)^n}$

How would I find the sum of this geometric series: $$ \frac{1}{1+x^2} + \frac{1}{(1+x^2)^2} + \dots + \frac{1}{(1+x^2)^n} $$ I want a formula, in the form of $\frac{n}{n+1}$, that can be proven by ...
4
votes
2answers
65 views

Find the maximum value of the fraction

Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b}<\frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime ...
2
votes
2answers
48 views

Proving by induction $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$

The context is as follows: I am asking this question because I would like feedback; I am a beginner to mathematical proofs. We wish to show $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + ...
2
votes
0answers
36 views

Why is it so hard to find a generating function for Somos' sequence?

The sequence is $\{1,2,12,576,1658880,\dots\}$. The $n$th number is obtained by squaring the $(n-1)$-th number and multiplying by $n$. So we start with $a_1=1$, $a_2=1^22=2$, $a_3=(1^22)^23=12$. In ...
0
votes
0answers
18 views

Partial sum of exponential series strictly increases after certain step

While trying to show that partial exponential series evaluated at two different values are strictly increasing provided that sufficient number of terms are applied I stuck at a problem. Given two ...
0
votes
3answers
39 views

Difficult nonlinear system based on max value

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 - xyz = 2$, $ y^3 - xyz = 6$, $ z^3 - xyz = 20$. The greatest possible value of $ a^3 + b^3 + c^3$ can be written in the form $ ...
0
votes
3answers
44 views

Does the following series converge uniformly?

I know how to show that the following series will converge absolutely. But am unsure how to show it will or will not converge uniformly for $z\in (0,1).$ $\displaystyle \sum_{n \mathop = 1}^{\infty} ...