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

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7 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
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0answers
9 views

Can the following product be written in the given form?

Can this: $$\eqalign{ & {\rm B}\left( {{m_3},{m_2} + m} \right)F\left( {m,{m_2};{m_3} + {m_2} + m;z} \right),where{\text{ }}{m_2},{m_3},z{\text{ are constant,}} \cr & {\text{m}} = ...
4
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1answer
82 views

a doubt with the series $ \sum_{n=0}^{\infty}e^{-nx} $

I know that the series is equal to $$ \sum_{n=0}^{\infty}e^{-nx}= \frac{1}{1-e^{-x}}$$ However, if I expand each exponential term into a Taylor series I get $$ \sum_{n=0}^{\infty}e^{-nx}= ...
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2answers
24 views

Compute infinite sum of a arithmetico-geometric series $\sum_{i=0}^{\infty} \frac{i}{2^i}$ [duplicate]

I am trying to compute the sum $\sum_{i=0}^{\infty} \frac{i}{2^i}$ which I know should be equal to $2$, but I cannot prove it. If I am not mistaken, it should be a arithmetico-geometric series ...
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0answers
21 views

relations between the root test and the ratio test

relations between the root test and the ratio test I know the theorem is correct if they are exist $$ \lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} \leq \lim\inf\limits_{n\rightarrow ...
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0answers
11 views

Prove that a certain sequence of polynomials is symmetric

Given $p_0(x,y,z)=1$, $p_{n+1}(x,y,z)=(xy+yz+zx)p_n(x,y,z+1)+z^2(p_n(x,y,z+1)-p_n(x,y,z))$. Prove that all $p_n(x,y,z)$ are symmetric polynomials.
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1answer
89 views

How prove this sequence $a_{n}=\sqrt{n}+\frac{1}{2}-\frac{1}{8\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)?$

let sequence $\{a_{n}\}$ such $$a_{1}=1,a_{n+1}=1+\dfrac{n}{a_{n}}$$ show that: $$a_{n}=\sqrt{n}+\dfrac{1}{2}-\dfrac{1}{8\sqrt{n}}+o\left(\dfrac{1}{\sqrt{n}}\right)?$$ This result is china student ...
3
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1answer
20 views

Bessel Equations Addition Formula

So, I'm considering yet another tricky proof involving Bessel Functions. Basically, I'm trying to figure out how the following is true: $$J_n(\alpha + \beta) = \sum_{m = -\infty}^\infty ...
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1answer
56 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
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1answer
17 views

If $r_n\to r$ and $s_n\to s$, then $(r \star s)_M/M \to rs$.

I was going to ask this question, but I think I figured it out, so I thought I'd post my answer: In this question of mine, a user's answer makes the following claim: Suppose $r_n$ and $s_n$ are ...
2
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2answers
32 views

Very slow convergence of a particular series?

I've read that $$ \sum_{k=2}^{\infty} \frac{1}{k (\log k)^2} = 2.1097\ldots $$ However when I compute the partial sums it looks like a lot of terms are needed to even get the first decimals right. My ...
2
votes
2answers
57 views

Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$

Does anyone know the sums of the following two series? $$\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$$ $$\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{4n-1}}{4n-1}$$ I encounter such series in my work.
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1answer
65 views

Is $(1+2+3+…)=(1+2+2^2+2^3+…)(1+3+3^2+…)(1+5+5^2+…)…$?

Are these equal? $$(1+2+3+…)=(1+2+2^2+…)(1+3+3^2+…)(1+5+5^2+…)…$$ Where the RHS has a series for each prime. Looks like they are the same series by the fundamental theorem of arithmetic. Every number ...
3
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2answers
25 views

Another Divergent Series Question

Suppose the series $$\sum{a_n}$$ diverges where $a_n\ge 0$ and the sequence is monotone non-increasing. If exactly one element is chosen from each interval of size $k$ -- i.e., one element from ...
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1answer
23 views

Divergent Infinite Series Question

If the infinite series $$\sum{a_n}$$ diverges where all terms are positive and $\lim{a_n}=0$, is it always possible to construct a subsequence such that its series converges? That is, does there ...
3
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0answers
52 views

Convergence of $\sum_n |\frac{\cos(3^n)}{n}|$

So a recent post asked about convergence of $\sum_n |\frac{\cos(2^n)}{n}|$, and using double-angle formula for $\cos$ it could be shown that for each pair of consecutive terms, at least one term had ...
2
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0answers
35 views

Understanding underlying algebra for calculus convergence problem

I'm working on series convergence/divergence problems in my Calc 2 class, and (as has happened often), I'm hung up on some underlying algebra. The first step in the solution manual for a problem I'm ...
2
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1answer
163 views

Does series converge or not?

$$\sum_{n=1}^\infty~\left|\frac{\cos2^n}{n}\right|$$ I just confused what to do.
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2answers
38 views

Uniform convergence of the series

Test the uniform convergence of the series $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$ $$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$ Can I find $M_n$ such that $$ ...
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0answers
31 views

sequence and series of convergence problem

We have a sequence/series problem. and i need help Consider the following: How can we show that $2-2^{1/2}+2^{1/3}-2^{1/4}+2^{1/5}-2^{1/6}+\ldots$ diverges? I am so lost as to solving this problem. ...
3
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1answer
50 views

Convergence of the series $\sum_{n=0}^\infty \frac{1}{n+1}\sin\bigr(\frac{p\pi u_n}{q}\bigl)$

Let $(u_n)_{n\in \mathbb{N}}$ defined by : $u_0=1, u_1=1$ and for all integer $u_{n+1}=3u_n-u_{n-1}$ Study the convergence of $$\displaystyle\sum_{n=0}^\infty ...
3
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1answer
55 views

Simple demonstration for $\lim_{n\to\infty}(\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}) = \frac{1}{e}$ [duplicate]

What is the simple demonstration with elementary means for Lalescu Sequence: $$\lim_{n\to\infty}(\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}) = \frac{1}{e}?$$ (Traian Lalescu-romanian mathematician (1882-1929))
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1answer
65 views

If $\sum\limits_na_n$ diverges and $(a_n)$ is positive decreasing then $\sum\limits_n\min(a_{n},\frac{1}{n})$ diverges [duplicate]

If the sequence $\{a_{n}\}$ monotonically decreases to $ 0$ and the series $\sum\limits_{n}a_{n}$ diverges, then the series $\sum\limits_{n}\min(a_{n},\frac{1}{n})$ diverges as well. my idea: ...
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0answers
21 views

how to generate a magic series?

Please explain the concept of magic series AND I want to generate a magic series for a given range of numbers. For instance, assume range = [0..10] for each index of the range I want a corresponding ...
2
votes
2answers
68 views

What is the sufficient and necessary condition for changing the order of summation?

What is the necessary and sufficient condition for $\sum\limits_{i=0}^{\infty }{\sum\limits_{j=0}^{\infty }{{{a}_{ij}}}}=\sum\limits_{j=0}^{\infty }{\sum\limits_{i=0}^{\infty }{{{a}_{ij}}}}$? Suppose ...
3
votes
1answer
29 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
2
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1answer
56 views

Hypergeometric Function simple identity

I must proove this property but I really have no idea of how to proove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z}) $$ It seems its a 'simple' property, but I haven't been able to ...
0
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1answer
29 views

Product of two geometric series

I have used the Product of two power series and find out the below results. But it is to some extend strange for me, could you please confirm the results? Let $A=\sum_{i=0}^{\infty}(\frac{L}{a})^i$ ...
2
votes
4answers
58 views

How to find answer to the sum of series $\sum_{n=1}^{\infty}\frac{n}{2^n} $

I have put his on wolfram and obtained answer as follows: $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$ And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$ However I am ...
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1answer
8 views

Insert Means in an Arithmetic Sequence (that contains logarithms)

So the question is: You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence Find 8 arithmetic means between them.
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2answers
20 views

Few questions about limit of sequence

Question:Find the the limits of these sequence.if the limits does no exist ,explain why. (1)$\left \{ cos((2n+1)\frac{\pi}{2}) \right \}_{n=1}^{\infty }$ my answer: so when n=1,lim=0 (2)$\left \{ ...
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1answer
34 views

How to find the Fourier series of a periodic function

Find the Fourier series of the function $f(t)=3t^2$, $-1\le t\le 1$. How do I solve this problem? What is the general formula and the way to solve this?
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3answers
60 views

How does this series diverge?

The series: $$\sum_{n=0}^{\infty} \sqrt{n^2 +1} -n$$ diverges. Can someone please tell me how this is proven and done.
9
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3answers
527 views

Example of a sequence with more than one limit.

I have heard of the idea of a sequence converging to more than one limit, but I cannot imagine how it would work. Could someone give me an example of such a case, and explain how it works?
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1answer
14 views

Convergent series b^(2n+2) for b < 1

For which $b$ ($<1$) is $$ \sum_{n=0}^{\infty}b^{2n+2} = 1 $$
2
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0answers
39 views

weak convergence and composition

Assume $X_n$ is a sequence of random variables defined on a common probability space and $X_n$ converges weakly (in distribution) to $X$ as $n \to \infty$. Assume $u_n$ is a sequence of integer valued ...
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1answer
13 views

Uniform Convergence of Series Help

Suupose the sequence $(b_k) , k\geq 0$ satisties $\sum k|b_k| < \infty$, then show that $\sum_{k=0}^\infty b_kx^k$ converges uniformly to a function $g$ on $|x| \leq 1$ and that $g'(x) = ...
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2answers
26 views

problems with sequence

consider the sequence $\{a_{n}\} _{n=1}^{\infty}$, $$a_n= \frac{1}{n^2}+\frac{2}{n^2}+\cdots+\frac{n}{n^2} $$ (1) find $a_{1}$, $a_2$, $a_3$, $a_4$' (2) by expressing $a_n$ in closed form, calculate ...
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1answer
11 views

What is the different between generating expression and infinite expansion?

What is the different between generating expression and infinite expansion? I can't figure it on my own
2
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0answers
32 views

Does this condition imply convergence of $\sum_k x_k$?

Let $\{x_n\}_{n \geq 1}$ be a sequence of real numbers (in principle, they could be complex numbers, but I don't think this makes much of a difference for this problem). Suppose the sequence $x_n$ ...
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3answers
37 views

Convergence of $\sum \frac{\sqrt{a_n}}{n^p}$

For $a_n \geq 0$, and $\sum a_n$ convergent, show that $\sum \frac{\sqrt{a_n}}{n^p}$ is also convergent for $p > 1/2$? What bugs me more is why isn't $\sum \sqrt{\frac{a_n}{n}}$ convergent?? ...
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3answers
40 views

Alternating p series. given that summation

Given that $$\sum_{k=1}^\infty{\frac{1}{k^2}} = \frac{\pi^2}{6}\ $$ Show that $$\sum_{k=1}^\infty{\frac{(-1)^{k+1}}{k^2}} = \frac{\pi^2}{12}\ $$
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4answers
57 views

Does the series converge or diverge?$\sum_{n=1}^\infty\frac{n^2-3n}{\sqrt[3]{n^{10}-4n^2}}$

Does this series converge or diverge? $$\sum_{n=1}^\infty\dfrac{n^2-3n}{\sqrt[3]{n^{10}-4n^2}}$$ I have tried using the comparison test, however when simplifying this i get $1/\sqrt[3]n$. I do not ...
2
votes
2answers
31 views

Find convergence or divergence of Series.

Determine which of the following series converge. Justify your answers. a) sum of (sin$(\frac{n \pi}{6}))^n$ b) sum of (sin$(\frac{n \pi}{7}))^n$ I believe that both sequences diverge because a sin ...
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votes
3answers
32 views

Limit of a exponential sequence

I am stuck with this tricky limit... Any idea? $$ \lim \limits_{n \to \infty}(n^2e^{-1/n}+ne^{-1/n}-n^2) $$ Of course, I am not allowed to use l'Hopital...
2
votes
1answer
23 views

Convergence almost everywhere

Let consider rational numbers $\{r_n\}_{n=1}^{\infty}$ on [0, 1]. How to prove, that such sum $$\sum_{n=1}^{\infty}\frac{1}{n^2|x-r_n|^{0.5}}$$ converges almost everywhere on [0, 1]. There are my ...
0
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2answers
26 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
0
votes
2answers
37 views

Calculate the following sequence $\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$

Calculate the following sequence $$\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$$
1
vote
1answer
44 views

Real Analysis question on sequences (Hint needed!!!)

Given the number $\alpha > 1$ , define the sequence an where $a_0 = 1$ and $a_{n+1} = (\alpha \times a_{n})^{\frac{1}{4}}$ for $ n \geq 0 $. Prove: If $a_{n}^{3}< \alpha $(as is true when n ...
1
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1answer
33 views

Prove that the following sequence converges and find its limit

Prove that the following sequence converges and find its limit $$\frac{1}{3+\frac{1}{3}},\frac{1}{3+\frac{1}{3+\frac{1}{3}}},\frac{1}{3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}},\ldots$$ I started to ...