All questions with tag [math: sequences-and-series]


Closed form of the sequence $a_{n+1}=a_n^2+1$

If $$a_{n+1}=a_n^2+1,$$ with first $a_1=\frac{1}{2}$. Just how to address this series trouble, i.e., how to stand for $a_n$ in shut kind?
2022-07-25 17:47:13

Maclaurin expansion of a given function

I am to expand $\ln(2+x)$ as a Maclaurin collection, I've obtained that $\ln(2+x)=\sum\limits_{n=1}^{ \infty}(-\frac{1}{2})^{n}x^{n}$. Can a person examine it?
2022-07-25 17:46:14

What is value of $\sum_{n=1}^{\infty}\frac{1}{(3n+1)^2}$?

Possible Duplicate: ยข Can you aid me? $$ \sum_{n=1}^{\infty}\frac{1}{(3n+1)^2} $$ or just how to represent it by various other methods?
2022-07-25 17:43:19

Alternating Series Test for Convergence - can the sequence be initially non-decreasing?

I was offered the adhering to question: Determine if the adhering to collection is convergent. You might make use of standard collection, yet you need to plainly state which results or rules you make use of. $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}\sqrt{n}}{n+4}$$ Does the Alternating Series Test call for the favorable term $\frac{\sqrt n}{n+...
2022-07-25 17:42:02

Studying the convergence of a series

I would love to research the merging of the collection: $$ \sum u_{n}$$ where $$ u_{n}=a^{s_{n}}$$ $$ s_{n}=\sum_{k=1}^n \frac{1}{k^b}$$ $$ a,b<1$$ We have: $$ u_{n}=\exp((\frac{n^{1-b}}{1-b}+o(n^{1-b}))\ln(a))=\exp(\frac{\ln(a)n^{1-b}}{1-b}+o(n^{1-b}))$$ However $$\exp(o(n^{1-b}))$$ has to be defined. So just how can I establish: $$ ...
2022-07-25 17:40:31

The Mandelbrot Set Membership

To define the Mandelbrot Set we consider a sequence of complex numbers $z_0$, $z_1$, $z_2$, $z_3$, with the following conditions: $$ \begin{cases} z_{n+1} &= &z_n^2 + c &\text{ for }n\geq 0, \\ z_0 &= &0. \end{cases} $$ where term $c$ is constant. Therefore, the sequence $z_0$, $z_1$, $z_2$, $z_3$ begins ...
2022-07-25 17:21:35

MA process ACF proof - don't understand it

I've got the proof but I don't understand a small detail. As you know for an MA process: $X_n = \sum _{i=0} ^q \beta_i Z_{n-i}$ where $Z_n$ is WGN (pure Gaussian random process). Then the ACF is: $\gamma(k) = Cov(\sum _{i=0} ^q \beta_i Z_{n-i}, \sum _{j=0} ^q \beta_j Z_{n-j + k}) = \sum _{i=0} ^q \sum _{j=0} ^q \beta_i \beta_j Cov(Z_{n-i}, Z_{n-...
2022-07-25 17:20:40

Use of the Reciprocal Fibonacci constant?

The Reciprocal Fibonacci constant ($\psi$) is specified as $$\psi=\sum_{k=1}^{\infty} \frac{1}{F_k}$$ where $F_{k}$ is the $k^{th}$ Fibonacci number. The unreason of $\psi$ has actually been confirmed. Does the Reciprocal Fibonacci constant have an usage in maths or is is remarkable merely due to the fact that it is the value of an intriguing...
2022-07-25 17:20:18

Finding a formula for the sum of a series that is neither Geometric nor Arithmetic

So I am offered a series where the terms $T(n)$ are: $1, 4, 11, 26, 57, 120$ and more. Each term is created by the amount $\sum\limits_{n=1}^k(2^n-1)$ I am being asked to share this amount over in regards to $n$ without the sigma symbols, such that I can create any kind of term $T(n)$ by connecting in a value of $n$ where $n$ comes from the in...
2022-07-25 17:18:28

Sub-sequences and Uniform Convergence.

Let $f_n$$^\infty_{n=1}$ be a sequence of functions defined on a set $S$. Suppose that there is a positive number $M$ such that |$f_n(x)$| $\leq M$ for every $n$ and for every $x \in S$. 1) If we choose any point $x \in S$ then show that there is a subsequence of functions $f_{n_k}$$^\infty_{k=1}$ such that it converges as $k \to \infty$. Is t...
2022-07-25 17:16:59

To prove $f(x)\rightarrow \infty$ with a "home made" strategy

My goal is to prove that: $\displaystyle\sum\limits_{n=1}^\infty \frac{1}{n^{x}} \rightarrow \infty$ for all $ x\rightarrow 1^+$ In order to show this statement I show that no matter how big you choose $N\in \mathbb{R}$, you can always find a $\delta >0$ so $\displaystyle\sum\limits_{n=1}^\infty \frac{1}{n^{x}} = 1+\frac{1}{2^{x}}+\cdo...
2022-07-25 17:16:04

Coefficient signs in the sum of successive powers of a polynomial

I'm searching for some structure in the sign variation of the coefficients of: $$P = \sum_{i>0} p^i\enspace,$$ for some polynomial $p \in \mathbb{Z}\langle x\rangle$ with no constant term. I'm interested in whether there is a simple algorithm which, given $n$, will tell me the sign of the coefficient of $x^n$ in $P$. In other words, I'm...
2022-07-25 17:13:59

Non-finite series implies product is zero

Given $0 \le y_n \le 1$ and $\sum_{n \in \mathbb{N}} y_n = \infty$, how can we show $\prod_{n=1}^\infty (1 - y_n) = 0$?
2022-07-25 17:07:58

How does the definition of "limit" capture the idea that a sequence gets "closer and closer" to the limit?

The series $x_n=2+1/n$ absolutely obtains "closer and also closer" to $2$ as $n$ obtains "larger and also larger." And we understand that $$\lim_{n\to\infty}2+1/n=2$$ What is taking place below? Make use of the definition to clarify what we actually suggest when we claim a series obtains "closer and also closer" to...
2022-07-25 17:06:45

To prove $f(x)\to\infty$ with an "Oresme" strategy

My goal is to prove that: $\displaystyle\sum\limits_{n=1}^\infty \frac{1}{n^{x}} \rightarrow \infty$ when $ x\rightarrow 1^+$ My first approach (which failed) is here: To prove $f(x)\rightarrow \infty$ with a "home made" strategy I think the confusing part is the "$x\rightarrow 1^+$". I have now argued for the statement, b...
2022-07-25 17:06:41

Do the notions of weak and weak* convergence coincide for $\ell^1(\mathbb{N})$?

As my friends and I were studying for our real analysis final exam yesterday, we were playing with various examples and found ourselves asking this question: The space $\ell^1(\mathbb{N})$ is the dual of $c_0(\mathbb{N})$, and the dual of $\ell^1(\mathbb{N})$ is $\ell^\infty(\mathbb{N})$. Is it possible to have a sequence $\{b_n\}\in\ell^1(\mat...
2022-07-25 17:02:12

Passing from induction to $\infty$

Somehow, the operation of passing to the limit after I have shown that something is true by induction for each natural number $n$ troubles me each time. I know there are instances where one cannot deduce a statement is true for $\infty$ if it holds for each $n \in \mathbb{N}$, and sometimes one can. Here is one instance where I am not sure wheth...
2022-07-25 16:59:21

Number of combinatorial progressions

A $k$-term combinatorial progression of order $2$ is defined as a set of positive integers $A=\{x_1<x_2<\cdots x_k\}$ such that the set $\{x_{i+1}-x_i:1\le i\le k-1\}$ has cardinality at most $2$. My question is given a positive integer $N$, how many $k$-term combinatorial progressions of order $2$ will exist in $\{1,2\cdots N\}$?...
2022-07-25 16:57:57

Compound interest - how to solve this with logarithms & geometric series?

I can make use of some aid with the following: Jacques is conserving for a new auto which will certainly set you back 29000dollars. He conserves by placing 400 bucks a month right into an interest-bearing account which offers 0.1% passion each month. After the amount of months will he have the ability to acquire his auto? (think it does not in...
2022-07-25 16:56:58

Predict next number from a series

Which approaches I can make use of to forecast next number from a collection of numbers? I recognize the minutes & max feasible number beforehand.
2022-07-25 16:56:36