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

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Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
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1answer
22 views

convergence of the series to inf or not

let $a_n = \dfrac{e^{-(1/2) \times a^2 \times\log(n) }}{a\sqrt{2\pi \log(n)}} $, $a$ is a constant, and the question is if $S_n = \sum a_n$ converge to a finite number. I wonder if I should ...
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2answers
42 views

Does alternating test show divergence?

My book states the alternating tests' convergence requirements. However, my book doesnt point out, if $a_n$ fails one of the convergence requirements, is it true that is diverges? Such as the limit ...
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0answers
48 views

Finding the convergent value of a recursion similar to Arithmetic-Geometric Mean recursion

The sequence is defined as follows : Start : $(x_0,y_0)$ with $ 0 < x_0 < y_0 $ Step : $x_{n+1} = \frac {x_n+y_n} {2}$ , $y_{n+1}= \sqrt{x_{n+1}y_n} $ Find $\lim_{n\to \infty}(x_n,y_n)$ . ...
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5answers
89 views

How to determine if the sequence $a_n= (30+12\arctan(n!))/6^n$ is divergent or convergent

$$a_n= \frac{30+12\arctan(n!)}{6^n}$$ Not sure where to start, I know at infinity arctangent tends towards $\frac{\pi}{2}$. I also know I'm supposed to find the limit but not sure how to start, the ...
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1answer
33 views

Interval of converge of $\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$

Find the interval of converge of: $$\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$$ I will use the ratio test. Let $\displaystyle a_n = \frac{n!(x+1)^n}{(2n-1)!}$ $\displaystyle a_{n+1} = ...
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2answers
45 views

Question about convergence of sequences

Given a sequence $x_n$ = $\sqrt{1}$ , $-\sqrt{1}$,$\sqrt{2}$,$-\sqrt{2}$... If $$y_n = \frac{x_1 + x_2 + x_3 + x_4 + \ldots +x_n}{n} $$ Then sequence $y_n$ is 1.Monotonic 2.NOT bounded ...
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2answers
40 views

Proving only the summation part of Cauchy-Schwarz

Can you prove only the summation part of Cauchy-Schwarz? What I mean is that $$\Bigl(\sum a_i b_i\Bigr)^2 \leq \sum a_i^2 \sum b_i^2.$$ I only want to show it for the case where $a_i , b_i \geq 0$ ...
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2answers
66 views

Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$

Evaluate: $$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$ Using real methods only. I am not sure what to do. I tried finding a power series, which was too ugly. I just need some ...
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1answer
37 views

Square Root of the Product of Convergent Series Converges

How do I go about proving that Square Root of the Product of Convergent Series Converges, where both are greater than 0. So $\sum \sqrt{x_n y_n}$, where $\sum x_n$ and $\sum y_n$ converge, and each ...
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1answer
38 views

Recursively appending mean to list: Is there a closed form?

I'm pondering the following sequence: $$\begin{equation} \begin{split} a_1 & = b \\ a_{n+1} & = c\frac{1}{n}\sum_{k=1}^{n}a_k = c \times \mathrm{mean\ of} \{a_1,\dots,a_n\} \end{split} ...
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1answer
5k views

How to determine if the series $\sum \frac{2+\sin n}{5^n}$ is convergent or divergent?

I have: the series $\sum_{n=0}^{\infty} \frac{(2+\sin n)}{5^n} $. I have split this up into $\sum_{n=0}^{\infty} \frac{2}{5^n} + \sum_{n=0}^{\infty} \frac{\sin n}{5^n}$. I know the first part is ...
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2answers
34 views

How to find the limit of the sequence

$X_0 := 2$ and for $X_n$: $X_{n+1} = \frac12 X_n + \frac1X_n$ I know that the sequence is monotone decreasing and I am not sure how to find its limit maybe with Cauchy and partial sums but still I ...
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2answers
303 views
+150

Integer part of a sum (floor)

Let $\left(\, x_{n}\,\right)_{\,n\ \geq\ 1}$ be a sequence defined as follows: $$ x_{1}={1 \over 2014}\quad\mbox{and}\quad x_{n + 1}=x_{n} + x_{n}^{2}\,, \qquad\forall\ n\ \geq\ 1 $$ Compute the ...
5
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1answer
123 views

Closed form formula for $2^{2^1}+2^{2^2}+…+2^{2^n}$

So, the series's sum is of below form: $2^{2^1}+2^{2^2}+...+2^{2^n}$ This series is an intermediate work of an bigger problem {So I am concerned with whether ...
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0answers
25 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
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2answers
852 views

Find the exact value of the infinite sum $\sum_{n=1}^\infty \{\mathrm{e}-(1+\frac1n)^{n}\}$

How can we find the exact value of the infinite sum $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$ This problem appears in: T. Andreescu, T. Radulescu & V. ...
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3answers
87 views

A series involving $\prod_1^n k^k$

Is this series $$\sum_{n\geq 1}\left(\prod_{k=1}^{n}k^k\right)^{\!-\frac{4}{n^2}} $$ convergent or divergent? My attempt was to use the comparison test, but I'm stuck at finding the behaviour of ...
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2answers
62 views

Questionable Power Series for $1/x$ about $x=0$

WolframAlpha states that The power series for $1/x$ about $x=0$ is: $$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$ This is supposedly incorrect, isnt it? This is showing the power series about ...
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0answers
40 views

How to prove that the series converges?

Let us suppose that $\vert a_n\vert$ is a real sequence, and set $S_n=\sum_{k=1}^{n}a_k$ and $\sigma_n=\frac{1}{n+1}\sum_{k=1}^{n}S_k$. How can we show that if the series $\sum_{n=1}^{\infty}\vert ...
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2answers
52 views

Prove two identities relating to series

Show that: $$ (1). \sum_{n=1}^{\infty}\ln\big(\cos \frac{x}{2^n}\big)=\ln \frac{\sin x}{x} $$ $$ (2). \sum_{n=1}^{\infty}\frac{1}{2^n}\tan \frac{x}{2^n}=\frac{1}{x}-\cot x $$ Thank you in advance. ...
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0answers
7 views

minimal polynomial of a LFSR sequence

I encircled the problem in the figure below. My question is, why $m(x)$ must have degree of at least $u$, why not it has a degree less than $u$? Maybe this is trivial, but I cannot wrap my mind ...
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4answers
40 views

Infinite Sum of 1/Polynomial

I'm trying to solve this equation: $$\sum_{k = 0}^{\infty}\dfrac{1}{(k+1)(k+3)}$$ Original image at http://i.imgur.com/wXZFxn0.png I attempted to find the sums of $\sum_0^∞\frac{1}{k+1}$ and ...
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2answers
41 views

If a sequence converges, then the sequence of its averages converges to the same limit [closed]

Assume $\{a_n\} \subset \mathbb{R}$ converges to $p \in \mathbb{R}$. Define $\{b_n\}$ by $b_n:=\frac{1}{n} \sum_{k=1}^na_k$. Prove that $\{b_n\}$ converges to $p$.
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1answer
36 views

Prove $f:\mathbb{N} \rightarrow \mathbb{R}$ is continuous using the definition of sequential continuity

The definition of sequential continuity is that $x_n \rightarrow x \implies f(x_n) \rightarrow f(x)$. If the terms of the sequence $\{x_n\}$ are only natural numbers, I know that for all $\epsilon ...
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3answers
67 views

Value of $\sum_{n=0}^{\infty} \exp(-bn^a)$

What is the value of $$ S = \sum_{n=0}^{\infty} \exp(-bn^a) $$ where $a>0$ and $b>0$? I know that $$ \int_0^{\infty} \exp(-bx^a) \, \mathrm{d}x=\frac{1}{ab^{1/a}}\Gamma(1/a) $$ but cannot ...
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2answers
16 views

Blocks of Pyramid Pattern Expression [duplicate]

There is a pattern following, and trying to find the algebraic expression Each layer (from the top). Diagram. So the first layer has 1, second has 4, third has 9, and the fourth has 16. That's ...
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1answer
36 views

Can someone help me prove that this sequence converges? [on hold]

I'm having difficulty trying to prove this, mostly because I don't understand the process. There was a proof similar to this in my textbook where they proved if Sn converges to s then lim 1/Sn=1/s and ...
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1answer
325 views

compound interest with geometric series

Were studying geometric sequences in maths and this came up as one of the questions: A mortgage is taken out for 150000 and is repaid annually with 20000 installments. Interest is charged on the ...
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1answer
11 views

Sequence Convergence when using a forgetting factor $\lambda$

I'd like to know how I can find the convergence formula of the following sequence $x_{i} = \lambda x_{i-1} + y$ with $\lambda \in (0,1)$, $y$ a positive scalar and initially starting from $0$, so ...
2
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2answers
54 views

let $(a_n)$ be a sequence of real numbers such that $|a_{n+1}-a_n|\leq \frac {n^2}{2^n}$ for all $n\in \mathbb N$. Then

let $(a_n)$ be a sequence of real numbers such that $|a_{n+1}-a_n|\leq \frac {n^2}{2^n}$ for all $n\in \mathbb N$. Then (a). The sequence $(a_n)$ may be unbounded. (b). The sequence ...
0
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1answer
17 views

Special case of limsup inequality

If you are given two sequences $a_{n}$ and $b_{n}$ such that $a_{n}b_{n} \geq 0$ and the limits $\lim\limits_{n \rightarrow \infty}a_{n} = a$ and $\lim\limits_{b \rightarrow \infty}b_{n} = b$ then can ...
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1answer
26 views

Count and description of vertices of certain faces of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$

For $k \ge 1$, $d \ge 2$ and $k \le d - 1$, let ${}^f_d\Omega^t_{d+k} (d;c_k(d - 1))$ be the intersection of $k - 1$ facets of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$ with equations: ...
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2answers
76 views

Series sum $\sum 1/(n^2+(n+1)^2)$

In an exercise, I caculate the Fourier expansion of $e^x$ over $[0,\pi]$ is $$e^x\sim \frac{e^\pi-1}{\pi}+\frac{2(e^\pi-1)}{\pi}\sum_{n=1}^\infty \frac{\cos ...
7
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1answer
56 views

$a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$

Suppose that $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$. I can find $\lim_{n \rightarrow \infty}a_n=0$. But I have no idea to find $\lim_{n \rightarrow ...
2
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2answers
45 views

How to compute the residue of $(z^2+2z+1)\sin\left(\frac{1}{1+z}\right)$

This was an example given in my notes but all it concluded was with something about an infinite principal part. How do we compute it? we have it equal to $ \left( z + 1 \right)^2 \cdot \sin \left( ...
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1answer
49 views

$\sin\left(1+\frac{1}{z-1}\right)$ expanded in powers of $z-1$

The whole problem: Obtain the following Laurent expansions. State the first four nonzero terms. State explicitly the $n$th term in the series, and state the largest possible annular domain in which ...
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1answer
15 views

The Laurent series expansion of 1/((z^2)(z+1))

In an example in our notes it says: Compute the Laurent series for f(z)= 1/z^2(z+1) and determine the annulus of convergence. No more information was provided. So I did it on my own by factoring it ...
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4answers
51 views

Power series for $f(x) = \frac{4}{x+2}$

Find the power series $f(x) = 4/(x+2)$ We know the geometric series: $$\sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x}$$ $(x+2) = 1 - (-x - 1)$ So: $$\sum_{n=1}^{\infty} (-1)^{n-1}\cdot(x + 1)^{n-1} ...
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1answer
34 views

Prove that if $\{a_n\}$ is a sequence of natural numbers such that $\{ a_n^{-1}\}$ is an arithmetic sequence then all $a_i$ are equal

Prove that if $\{a_n\}$ is a sequence of natural numbers such that $\left\{\frac{1}{a_n}\right\}$ is an arithmetic sequence then all the $a_n$s are equal. I have no clue where to begin from, ...
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1answer
27 views

Series convergence issues

I want to show that $$\sum_{k=0}^\infty \frac{1}{1+x^n}$$ is $C^1$ on ($1,\infty)$ To do so, I want to show uniform convergence of $\sum_{k=0}^\infty \frac{d}{dx}\frac{1}{1+x^n} = \sum_{k=0}^\infty ...
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0answers
46 views

How find this sequence recursive relations

Question: Let $$A_{n}=\sum_{i=0}^{n-3}(-1)^{n+i-2}\dfrac{13n^2-31n-10ni+9i+i^2+16}{(3n-i-3)(3n-i-4)(2n-i-3)!\cdot i!}$$ I want find the $A_{n}$ recursive relations,such as following form ...
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1answer
34 views

Convergence of a sequence involving integral

Consider $f:[-\pi, \pi] \to \mathbb{C}$ is analytic (infinite differentiable) and periodic. Define $a_n:= \frac{1}{2 \pi}\int_{-\pi}^\pi f(x) e^{-inx}dx$ (the Fourier coefficient of $f$). Show ...
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1answer
37 views

How to evaluate this infinite sum

I want to find $\int_0^1(1-x)^{\frac{1}{3}}x^{1/3}dx$. From binomial theorem, $(1-x)^{\frac{1}{3}}= \sum_{0}^\infty (-x)^n\binom{\frac{2}{3}}{n}$. Then $\int_0^1(1-x)^{\frac{1}{3}}x^{1/3}dx= ...
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1answer
18 views

If a,b,c are in AP and $a^2,b^2,c^2$ are in HP, then prove either $a=b=c$ or $a,b,- \frac c2 $ are in GP

As the title says. Although first part of the proof is obvious, I'm still able to prove it. And for the second part, I'm essentially trying to prove $b^2=-c/a$ (which is possible only when c<0 ...
0
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1answer
48 views

Problems on sequence and series of functions

Let $a_n$ be a sequence of real numbers. Which of the following is true? a. If $\sum a_n$ converges,then so does $\sum a_n ^4.$ b.If $\sum |a_n|$ converges,then so does $\sum a_n ^2.$ ...
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0answers
39 views

Evaluating $\sum \limits_{i=1}^{\infty} i^2 \exp\left[- \frac{(i+1/2)^2}{2s^2}\right] , \ s>0 $

How can we evaluate the following sum. $$ \sum \limits_{i=1}^{\infty} i^2 \exp\left[- \frac{(i+1/2)^2}{2s^2}\right] , \quad s>0 $$
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4answers
97 views

On limits related to $\frac1{\ln n}\sum\limits_{r=1}^{n^4}\frac1r$ when $n\to\infty$

$$L=\lim_{n\to\infty}\frac1{\ln n}\sum_{r=1}^{n^4}\frac1r,\qquad M=\lim_{n\to\infty}\left\lfloor\frac1{\ln n}\sum_{r=1}^{n^4}\frac1r\right\rfloor$$ I know that ...
1
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2answers
33 views

I can't understand this difference equation step

I am working on birth-death processes and I can't understand a step that is taken in a proof. The mean of a process is defined as $$\mu(t) = \sum_{n=1}^{\infty}np_n(t)$$ At certain stage in the ...
0
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1answer
33 views

What is the difference between the limit of a sequence and a limit point of a set?

I always thought they were the same thing. The limit of a sequence is a point such that every neighborhood around it contains infinitely many terms of the sequence. The limit point of a set is a point ...