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

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0answers
15 views

Power Series Representation of $x^3/(2-x)^3$

I don't need an answer, as this was a question I got wrong on a problem set, but could someone explain this? So, we have to represent f(x)= $x^3$/$(2-x)^3$ My professor writes consider g(x) = ...
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3answers
31 views

How do you add two series together

How do you add the series $$\frac{1}{2}(\sum_{n=0}^{\infty}\frac{2^{n}}{(z-3)^{n+1}} + \sum_{n=0}^{\infty}\frac{(z-3)^{n}}{4^{n+1}})$$ ? is this right? ...
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1answer
21 views

Pointwise convergence of $c$-concave envelopes as the sequence of cost functions converges pointwise

On Remark 1.12, page 33 of Villani's Topics in optimal transport we find the following problem. It should be easy but it's turning me mad. Let $c:\mathbb{R}^n\times\mathbb{R}^n\rightarrow ...
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1answer
21 views

using the ratio to prove convergence

Let $(a_n)_{n \in \mathbb{N}}$ be a complex sequence and assume \begin{equation*} \exists N\in \mathbb{N},\alpha>0~\forall n\geq N: a_n\neq 0,~|\frac{a_{n+1}}{a_n}|\leq ...
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0answers
21 views

Uniform convergence to a differentiable function

Let $(a_n)_{n\in \mathbb{N}}$ be a bounded sequence. Prove that the series $\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{n^{2x}}$ converges absolutely and uniformly on $(1, +\infty)$ to a ...
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0answers
25 views

Comparing ordered numerical sets by their density

Cantor's idea of comparing infinite sets by making one to one correspondences between their members seems to be very fruitful. Yet I wonder whether anybody ever suggested to compare ordered and ...
3
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2answers
36 views

Radius and interval of convergence of $\sum_{n=1}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$ by root and ratio test are different?

$$\sum_{n=1}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$$ By using ratio test $$\lim_{n\to\infty}\frac{x^{2(n+1)}}{(2(n+1))!}\frac{(2n)!}{x^{2n}}=\lim_{n\to\infty}\frac{x^2}{(2n+2)(2n+1)}=0$$ By using root ...
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3answers
22 views

Finding two sequences with a limsup value

Find two sequences ⟨ X n ⟩ and ⟨ Y n ⟩ such that limsup(Xn + Yn )=E=Euler's constant and limsup Xn + limsup Yn =Pi=π . I couldn't ...
2
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1answer
52 views

Question regarding Fourier Series

Things I understand (scroll down to see question in bold): Let $T$ be the function's period Let $w_0 = \frac{2π}{T}$ A function $x(t)$ can be written as the sum of its even and odd parts, that is ...
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0answers
16 views

Closed form for sum resembling generating function

Is there a general closed-form solution for $$\sum_{k=0}^\infty \frac{f(k)}{z-k}$$ as a generating function of $f$? It vaguely reminds of a couple of other kinds of generating functions, but not in ...
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3answers
45 views

A problem on convergent subsequences

Find a sequence $\langle X_n\rangle$ such that $$L_{X_n}=\left\{\frac{n+1}{n}:n\in\mathbb N\right\}\cup \{1\}.$$ Where $L_{X_n} = \{ p\in \mathbb{R} : \text{There exists a subsequence } \langle ...
2
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0answers
24 views

Proof on Riemann's Theorem that any conditionally convergent series can be rearranged to yield a series which converges to any given sum.

I am looking at the proof of the following theorem from Apostol's Mathematical Analysis. I am having trouble showing the last part that the author left to the reader. I'm trying to show that $y$ is ...
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2answers
70 views

Find $\sum_{n \ge 1} 1/n^2$ using the Fourier expansion of $f(x) = x$

The strategy I have been asked to take, is to show that Fourier coefficients of the function $f(x) = x$ on $[0, 1]$ are up to a constant equal to $1/n^2$.Then I should apply the norm ...
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1answer
45 views

Challenging Closed form Questions

This is a rather odd request. I was given a task to create some challenging (as much as possible) problems on Closed Form questions and also provide solutions to them. Could somebody please suggest ...
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2answers
44 views

Prove that this sequence diverges [on hold]

Prove that this sum diverges: $$\sum_{n=1}^\infty (\frac{1}{n})^\frac{1}{n}$$ as $n\to\infty$
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3answers
28 views

convergence, finding limit

I just came across an exercise, however I don't know how to find the limit of $$\lim_{n \to \infty} \frac{2^n}{n!}$$ can any body help? Of course this is not homework, I'm only trying out example ...
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1answer
49 views

Compute $\lim\limits_{n\to \infty}{\prod\limits_{k=1}^{n}a_k}$ Where $a_k=\sum_{m=1}^{k} \frac{1}{\sqrt{k^2+m}}$

I showed that: $$\frac{1}{\sqrt{1+{1 \over {k}}}} \leq a_k \leq \frac{1}{\sqrt{1+{1 \over {k^2}}}}$$ And then $$\lim\limits_{n\to \infty}{\prod\limits_{k=1}^{n}a_k} \leq \lim\limits_{n\to ...
1
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1answer
35 views

Number of “rising/increasing sets”?

I have the following problem to solve: a) was pretty easy to show, but I am struggling to count the sequences in b. So far I noticed the obvious: $$|T_{i}|\geq i$$ Counting the sequences leads to a ...
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1answer
30 views

Find explicit formula for nth term in series [on hold]

find an explicit formula for the nth term, and the sum of the first nth terms for the following sequence: 1, 1, 2, 2, 4, 4, 8, 8, 16, 16, ... thanks
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0answers
22 views

Limit Comparison Test for negative sequences

So I have a $\sum a_n < \infty$ and $a_n>0$. We also know that $\lim_{n\to\infty} b_n = 0$. I need to show that $\sum a_n b_n < \infty$. For $b_n>0$, it's easy to show that using the Limit ...
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votes
2answers
29 views

${a_n}$ for a sequence containing no zeroes

Take the sequence of Natural numbers which do not contain the digit zero. So your sequence becomes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12 ... Can we find an expression for ${a_n}$ ?
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5answers
50 views

Formula for $r+2r^2+3r^3+…+nr^n$ [duplicate]

Is there a formula to get $r+2r^2+3r^3+\dots+nr^n$ provided that $|r|<1$? This seems like the geometric "sum" $r+r^2+\dots+r^n$ so I guess that we have to use some kind of trick to get it, but I ...
0
votes
1answer
11 views

absolute convergence

I am trying to figure out of the series $$\sum_{n=1}^\infty \frac{-1^n}{n}-\frac{-1^{n+1}}{n+1}$$ converges absolutely. I know that we use the ratio test to see if a series converges absolutely, but ...
2
votes
1answer
22 views

Kempner Series with bases other than 10

Although the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges, we know that if we remove from the sum all the terms whose denominator expressed in base 10 contains a 9 digit, the ...
1
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1answer
19 views

Convergence of series: determining limit

I'm self-studying analysis from some notes that a university here have put up, and it contains a great many tests and criterias for determining when a series $\sum a_n$, or $\sum f_n(x)$, is ...
3
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2answers
382 views

What is $\lim_{n\to\infty}\displaystyle\sum_{r=1}^{n}\frac{1}{(r+2)r!}\:$?

$$\lim_{n\to\infty}\displaystyle\sum_{r=1}^{n}\frac{1}{(r+2)r!}$$ if we can find sum of this series then we can evaluate limit \begin{equation*} \frac{1}{(3)1!}+ \frac{1}{(4)2!}+\frac{1}{(5)3!}+ ...
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votes
1answer
36 views

convergence on sequences using cauchy's criterion [duplicate]

I have the following sequence $$x_{n+1}=\frac12\left(x_n+\frac2{x_n}\right),~n>1,~x_1=1$$ How can I prove the sequence is Cauchy, and get the limit?
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4answers
47 views

How can I convert the function $ g(x) = {1 \over (1-x)^2} $ to a power series and find the interval of convergence?

$$ g(x) = {1 \over (1-x)^2} $$ I know that a power series converges to $1 \over 1-x$ if $|x|<1$ But I'm confused by the exponent in the denominator.
1
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1answer
21 views

Laurent series , function representation

Write the Laurent series around zero for the entire function $f(z)=z^2e^{3z}$ I'm a little confused on how to represent the complex functions by series, as I did in the calculation of real functions, ...
0
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1answer
18 views

How to prove or disprove this statement about convergent sub-sequences? [duplicate]

I need to prove through a proof or disprove through a counter example that if $a_n$ is a sequence which has three sub-limits ( $a_n$ has three sub-sequences that each one of them have different ...
0
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4answers
30 views

Finding the last value of a quadratic sequence

$$\large 2,6,12,20,30, \ldots, a_n $$ Above shows the first few numbers of an increasing series. If the sum of these numbers equals $32430$, find the value of $a_n$. My progress. I defined the ...
0
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1answer
31 views

How can I find the convergence/divergence of $\sum_{n=1}^\infty (-1)^n{1 \over 2n+1}$?

$$\sum_{n=1}^\infty (-1)^n{1 \over 2n+1} = \sum_{n=1}^\infty a_n $$ When testing for absolute convergence I used the n-th term and ratio test, and got inconclusive for both using: $$ |a_n| = {1 \over ...
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2answers
24 views

Uniform convergence of $x\arctan(nx)$

$f_n(x)=x\arctan(nx)$; $n>=1$; $x$ belongs to $\mathbb R$. Prove that $f_n(x)$ uniformly convergent to a function $f$. I have proved that $f_n(x)$ point wise convergent to $f(x)=(\pi/2)|x|$. Now I ...
1
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1answer
20 views

Uniform convergence of $ x\arctan(nx)$

Let $f_n(x)=x\arctan(nx)$ for $n\geq1$. Show that $(f_n(x))$ converges uniformly to a function $f$. In the previous parts I have proved that $f_n(x)$ continuously differentiable and it converges ...
0
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0answers
23 views

The sum of the first $n$ terms of a Geometric Progression is $\frac43\left(4^n-1\right)$ Find its $n$th term as an integral power of $2$.

The sum of the first $n$ terms of a Geometric Progression is $$\frac43\left(4^n-1\right)$$ Find its $n$th term as an integral power of $2$. I know the formula of the $n$th term of a GP ...
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0answers
41 views

What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
2
votes
1answer
51 views

For which $z \in \Bbb{C}$ does this series converge?

How can I determine for which $z\in \Bbb{C}$ the following series converges? $$\sum_{n=0}^{\infty} \frac{z^n}{1+z^{2n}}$$ I've tried the root and ratio test with no success.
0
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1answer
37 views

How to find this sum: $\sum_{n=1}^{\infty}\frac{H^{(r)}(n)}{2^n n}$

I am learning new sums and I would appreciate your hints about how to approach the following $$S(r)=\sum_{n=1}^{\infty}\frac{H^{(r)}(n)}{2^n n}$$ where $$H^{(r)}(n)=\sum_{j=1}^{n}\frac{1}{j^r}$$ is ...
2
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1answer
31 views

Verification of Solution for Walter Rudin Principles of Mathematical Analysis Exercise 20, Chapter 3

I have written an answer for the problem 20, chapter 3 of Walter Rudin's Principle of Mathematical Analysis. I think the proof is correct, but since I am new with this kind of proofs, I am skeptical ...
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0answers
24 views

how to integrate this equation with two sums inside?

i'm reading a book, and i have trouble with this problem, i don't how to integrate this equation and where to begin from. Although they already give the answer but I don't understand how to get it. I ...
1
vote
1answer
25 views

prove or disprove this theorem about subsequences

I need to prove or disprove that for a certain sequence $a_n$, if the subsequence of the even indices and the subsequence of the odd indices are Cauchy sequences then $a_n$ converges.
3
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2answers
61 views

Interesting fact about convergent sequences?

Let $a_n > 0$ for $n=1,2,...,$ with $\sum_{n=1}^{\infty}a_n < \infty$. Prove that $b_n$ $(n=1,2,...)$ exist such that $b_n/a_n \rightarrow \infty$ as $n \rightarrow \infty$, but ...
0
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0answers
31 views

Evaluate $\lim\limits_{n \to \infty}a_n$ for a recursive sequence $a_{n+2}=f_{n \pmod 3}(a_{n+1},a_n)$ where $f_{k}(x,y)$ is the mean of $x,y$.

Prove the limit $\lim\limits_{n \to \infty}a_n$ exists and evaluate it for a recursive sequence $a_{n+2}=f_{(n\pmod 3)}(a_{n+1},a_n)$ where: $f_{0}(x,y)={1 \over 2}(x+y)$ $f_{1}(x,y)=\sqrt{xy}$ ...
2
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4answers
73 views

The convergence of the arithmetic mean into the geometric mean

Given $p$ positives values $a_1...a_p$, define the sequence $x_n$ such that: $$x_n = \frac{\sqrt[n]{a_1}+...+\sqrt[n]{a_p}}{p}$$ And define $S_n = (x_n)^n$ Prove that $S_n \rightarrow ...
1
vote
2answers
46 views

How to prove this statement using Cauchy's statement?

If $b_n>0$ is a sequence which implies that for every $\epsilon>0$ exists a certain $N$ so that for $m>n>N$ the expression $\sum_{k=n}^m b_k$< $\epsilon$ is true.It is also known that ...
3
votes
4answers
182 views

Recursive sequence with square root

I came across this (cool) question this weekend Find the limit of the following sequence as $n$ approaches infinity. $x_1 = 1$ and $x_{n+1} = \sqrt{x_n^2+\frac{1}{2}^n}$ I had two questions about ...
1
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2answers
28 views

Prove that every convergent sequence has a monotone subsequence [duplicate]

So if a certain sequence $a_n$ is convergent then its bounded.So from Bolzano-Weierstrass $a_n$ has a convergent sub-sequence, but where do I continue from here?
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votes
3answers
55 views

Sum of series $\sin \theta+\sin 2 \theta+\sin 3\theta+\dots$ [duplicate]

I need to prove sum of series: $$\sin \theta+\sin 2 \theta+\sin 3\theta+\dots=\sum_{n=1}^\infty \sin n\theta$$ by using in the first place the complex numbers.
0
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1answer
15 views

The limits of recursive sequences of different types of means - my solution + challenge

Consider the sequence $a_{n+2}=f(a_1,a_2)$ where $f(x,y)$ is the mean of $x, y$ (geometric/arithmetic/harmonic) and $a_1,a_2$ are positive real numbers. In detail: Geometric - ...
2
votes
3answers
48 views

Differentiation method for evaluating $ \sum_{n=1}^\infty \frac{n^2}{3^n} $

I evaluated the following infinite sum (the original and broader question regarding this sum can be found at Evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n} $). $$ \sum_{n=1}^\infty \frac{n^2}{3^n} $$ ...