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

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0
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0answers
10 views

Expanding sum of negative power using log

Example expands the series with natural logarithm but I do not know this rules. $\sum_{a} a^{-n} = \sum_{a}1 - n \sum_{a}log(a) + \cdots $ Anybody understand this expansion? Thank you!
2
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2answers
22 views

Does the series $\sum_{n=0}^{+\infty} \sin((1+\sqrt{2})^n\pi)$ converge?

I am trying the following exercise, Convergence of the series $\sum_{n=0}^{+\infty} \sin((1+\sqrt{2})^n\pi)$ I tried like the method for $\sum_{n=0}^{+\infty} \sin((2+\sqrt{3})^n\pi)$, with ...
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2answers
20 views

How to find the boundaries of a sequence

If $a(n)=\frac{-1}{n!}$ , how does one find the numerical boundaries of this sequence , rigurously ?
2
votes
1answer
46 views

Number of palindromic numbers less than a power of $10$

I noticed that every $10^{n}$ there is a certain number of palindromic numbers that I collected in this sequence: $$S=\{a_n,a_{n+1},a_{n+2}...\}=\{10,9,90,90,900,900...\}$$ where every number $a_n$ is ...
2
votes
2answers
90 views

Extremely hard sum of infinite series problem

I found this in a previous years iit paper. It seems really interesting, but incredibly difficult. Can someone please help $$ \sum_{i = 1}^\infty \arctan\frac{1}{2i^2} $$ This whole ...
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1answer
12 views

Question about the use of convergence tests

I just found out about convergence tests , found them on Wikipedia and I've got this question : Are you allowed to use , for example , the ratio test if your sequence is NOT defined as a sum ? Let's ...
0
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0answers
9 views

Question regarding the sum of the reciprocal of the values in Sylvester's sequence

The unit fractions formed by the reciprocals of the values in Sylvester's sequence generate an infinite series: $\sum_{i=0}^{\infty} \frac1{s_i} = \frac12 + \frac13 + \frac17 + \frac1{43} + ...
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0answers
40 views

Limit of a factorial sequence

Let $$a(n)=-\frac{1}{n!},$$ where $n\ge0$ . How do I prove the convergence of this sequence ? I am not allowed to use the ratio test for this sequence, since this is not a sum, right? So, how do I go ...
2
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1answer
29 views

Asymptotic behaviour of $\frac{\Gamma(n)}{\Gamma(n+\frac{6}{5}+i\frac{2}{7})}$

Find the asymptotic behaviour of $$\frac{\Gamma(n)}{\Gamma(n+\frac{6}{5}+i\frac{2}{7})}\ \ \ (n\rightarrow \infty)$$ I know we must use Stirling's formula. But I can't .Thank you
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0answers
17 views

Determing sequence from its Dirichlet series

Suppose I know the Dirchlet series $$\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \frac{\zeta(s)}{\zeta(3s)},$$ where $\zeta(s)$ is the usual Riemann zeta function. My question is - is there a way to ...
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1answer
20 views

Minimisation of Finite sum of a decreasing sequence

If $a_{1}<a_{2}<a_{3}<...<a_{n}$, find the minimum value of $$\sum_{i=1}^{n} (x-a_i)^{2}$$ Then find the value of $$f(x)=\sum_{i=1}^{n} |x-a_i|$$ Hi all, what would the best way be ...
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1answer
28 views

Prove $\sum^{\infty}_{n=k} (-1)^{n} \frac 1 {n^r} \le |\frac 1 {k^r}|$ for a natural number $r \ge 1$

Prove $\sum^{\infty}_{n=k} (-1)^{n} \frac 1 {n^r} \le |\frac 1 {k^r}|$ for a natural number $r \ge 1$ I know the above series converge since $a_n > a_{n+1}>0$. I have $\sum^{k}_{n=k} ...
0
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1answer
31 views

Can this sequence be expressed with a formula?

Can this sequence be expressed with a formula? 1 1 1 2 2 3 3 3 4 4 4 5 5 6 6 6 7 7 8 8 8 9 9 9 10 10 11 11 11 12 12 12 13 13 14 14 14 15 15 16 16 16 17 17 17 18 18 ...
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2answers
37 views

Show that $\sum\limits^{\infty}_{n=1} (-1)^n \frac {x^2+n} {n^2}$, $x \in \mathbb R$, converges uniformly on every bounded interval

Show that $\sum\limits^{\infty}_{n=1} (-1)^n \frac {x^2+n} {n^2}, x \in \mathbb R$ converge uniformly on every bounded interval $I \subseteq \mathbb R$. I've already shown that the series does ...
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0answers
26 views

Find an analytic function [duplicate]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that: 1) $f(0) = 0$ 2) $f(x_1) = y_1$ 3) $f(x) > 0, \forall x>0$ 4) ...
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5answers
182 views

Prove that $\sqrt{n} > \ln n$

Prove that $\sqrt{n} > \ln n$ for all $n \in \mathbb{N}$. I need to use this fact for one of the proofs that I am working on. However, I am having trouble proving this. I tried induction but don't ...
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2answers
16 views

A question about series ratio test

Could you please give me some hint how to deal with this question: Suppose $\left|\frac {a_{n+1}}{a_n}\right|\le c_n$ for each n and $c_n<1$. May we conclude that $\left|\frac ...
3
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1answer
25 views

Convergence of $\sum\limits_{n=2}^ \infty (-1)^n \frac{(\ln n)^p}{n^q}$ with $0<q < p$

Let $p$ and $q$ be positive real numbers such that $q < p$ , then is the following series convergent? $$ \sum\limits_{n=2}^\infty(-1)^n\frac{(\ln n)^p}{n^q} $$
3
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0answers
47 views

L'Hopital quicky

suppose L'Hopital applies and $$\lim_{x\to\infty}\frac{f(x)}{g(x)} = \lim\frac{f'(x)}{g'(x)}$$ under what conditions is it true then that $$\lim_{x\to\infty}\frac{\frac{f(x)}{g(x)} }{ ...
0
votes
2answers
19 views

Prove that the convergence of the sequence (s3n) implies the convergence of (sn).

I write $s_n-s$, as $(s_n^3-s^3)/(s_n^2+s_n*s+s^2)$, true for all $n>N$. I'm trying to show that the denominator is convergent. But I don't know how to do this. Need help! Thanks. (Sorry about ...
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1answer
23 views

Prove that the convergence of the sequence (sn) implies the convergence of (s3n) [on hold]

Case 1: $s>0$. Assume $s>0$. Then there exists $N$ such that for all $n>N$ $s_n>0$. If $s_n^3-s^3$ converges to $0$, then write $s_n-s$, as was done in OH, as ...
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0answers
22 views

Find the Intervel of Convergence of the Power Series

One question for you guys. I need to find the interval of convergence of the power series. Having a lot of problems with this one and would love a thorough explanation, however any help is ...
2
votes
2answers
32 views

The sum of reciprocal squares: estimating the remainder

Let $a_n$ denote the $n$th remainder of the series $$ 1+\frac{1}{2^2}+\frac{1}{3^2}+\ldots $$ In other words, $$ a_n = \frac{\pi^2}{6}-\left(1+\frac{1}{2^2}+\ldots +\frac{1}{n^2}\right). $$ I ...
0
votes
1answer
34 views

Is there name to the following sequence: $c_n = c_1c_2…c_{n-1} + 1$

I just saw the sequence $c_n = c_0c_1c_2...c_{n-1} + 1$ and is thinking whether sequence $(c_n)$ has some name. Add: What if $c_0 \neq 2$?
2
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0answers
33 views

What is this sequence of polynomials?

NovaDenizen says the polynomial sequence i wanted to know about has these two recurrence relations (1) $p_n(x+1) = \sum_{i=0}^{n} (x+1)^{n-i}p_i(x)$ (2) $p_{n+1}(x) = \sum_{i=1}^{x} ip_n(i)$ == i ...
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0answers
11 views

How to simultaneously search multiple finite sequences in OEIS?

Suppose I have several finite sequences and I want to know if they show up in a family of sequences or any one particular sequence or on rows of an array on OEIS; is there such a search option.
3
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5answers
93 views

$\sum\limits_{n=1}^\infty n(\frac{1}{2})^{n}$ [duplicate]

I am trying to find the expected value of the number of even numbers rolled before the first odd number when rolling a fair die until an odd number comes up. I arrived at $\sum\limits_{n=1}^\infty ...
1
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2answers
30 views

Prove $\lim_{n \to \infty} \sqrt[k]{X_n} = \sqrt[k]{\lim_{n \to \infty} X_n}$, where $\{X_n\}_{n=1}^\infty$ converges

Let $\{X_n\}_{n=1}^\infty$ be a convergence sequence such that $X_n \geq 0$ and $k \in \mathbb{N}$. Then $$ \lim_{n \to \infty} \sqrt[k]{X_n} = \sqrt[k]{\lim_{n \to \infty} X_n}. $$ Can someone help ...
0
votes
1answer
27 views

Prove existence of $\{X_n\}_{n=1}^\infty$, $\{Y_n\}_{n=1}^\infty \subseteq S \subseteq \mathbb{R}$, $S \neq \emptyset$ and $S$ bounded

Let $S$ contained in $\mathbb{R}$ be a nonempty bounded set. Then there exists monotone sequences $\{X_n\}_{n=1}^\infty$ and $\{Y_n\}_{n=1}^\infty$ such that $X_n$ and $Y_n$ is contained in $S$. How ...
0
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2answers
55 views

Find the sum of the following series

Find the sum of the following series $$ \sum_{n=1}^\infty (-1) \frac{1}{n}\frac{9}{6^n}. $$ I think that $r$ is $\frac{9}{6^n}$ and $a$ is $-1$. But I'm not positive if I'm starting this problem ...
-1
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0answers
37 views

Calculated by multiplying the arithmetic progression terms

$$\prod_{k=1}^n(a+(k-1)d)=a\cdot(a+d)\cdot(a+2d)\cdot(a+3d)\cdots(a+(n-1)d)=\text{?}$$ Please help me! What is the formula?
3
votes
3answers
118 views

Help with radius of convergence of a power series.

I need to determine the radius of convergence of the series $\sum_{n=1}^\infty a_nx^n$, where $a_n=a^n+b^n$ and $a,b$ are real numbers. Not sure how to approach this one.
-2
votes
1answer
21 views

Finding sum of special series [on hold]

How can you find the sum of this series. I got it in my test today, but I can't solve it (2^2/1*2)C0 + (2^3/2*3)C1 + .... + (2^12/11*12)C10 Where C0,C1... Are binomial coefficients
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votes
1answer
23 views

Explanation for sum of sequence

I saw that in a textbook. Could somebody explain how this sum of a sequence was obtained? ⌈n/2⌉+...+⌈n/2⌉+⌈n/2⌉ = ⌈(n+1)/2⌉⌈n/2⌉
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0answers
14 views

Combining even and odd parts of a Chebyshev series

I imagine this will be an easy problem, perhaps even routine, for some. I am learning to manipulate sums and need insight. I started with a power series $$s(x) = \sum_{n=0}^{\infty} a_n x^n$$ and ...
5
votes
1answer
43 views

Sequence Limit Problem: If $0 \leq x_{m+n} \leq x_n + x_m$ then limit of $x_n/n$ exists

If the sequence $\{x_n\}$ satisfies the property that $0 \leq x_{m+n} \leq x_n + x_m$ for all $n$, $m \in \mathbb{N}$ , show that the limit of the sequence $\left\{\frac{x_n}{n}\right\}_n$ exists. ...
4
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1answer
25 views

Divergent subsequence of an unbounded sequence

Let $(a_n)$ be a sequence of real numbers that is unbounded above. Show that $\exists$ a subsequence $(a_{n_k})_{k \ge 1}$ such that $\lim_{k \rightarrow \infty} a_{n_k} = + \infty$. Working ...
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2answers
22 views

General Term of specific Alternating Series

Recently I think about series below and I wonder if there is away to write the general term of it... $$ ...
6
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0answers
74 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
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1answer
19 views

How to make a formula to link two sequences without any positioning numbers?

I have two sequences: sequence $n$: $4, 7, 10, 13$ sequence $p: 10, 16, 22, 28$ I wish to find a link between them i.e. $p=??$ However, I cannot use positioning numbers, so you can't say that $4$ ...
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vote
4answers
65 views

Why do we assign values to divergent series? [duplicate]

Why do we assign values to divergent series? For example, the series $1+2+3+4... = -1/12$. I understand the proof for this, but I feel like it uses false math, and I recall reading that you can't do ...
0
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0answers
18 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
11 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
votes
1answer
90 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}= ...
1
vote
1answer
26 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 ...
0
votes
0answers
25 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 ...
0
votes
0answers
12 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.
8
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1answer
147 views
+50

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
votes
1answer
42 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 ...
1
vote
1answer
60 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.