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

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0answers
23 views

To construct a power series such that the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n b_n x^n$ is $2R$.

Let $\sum_{n=0}^{\infty} a_n x^n$ is a power series with radius of convergence $R(>0)$. To construct a power series $\sum_{n=0}^{\infty} b_n x^n$, other than $\sum_{n=0}^{\infty} (\frac x2)^n$, ...
2
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2answers
30 views

Asymptotic Expansion of $\ f(x)=(1-\beta \frac{ log(log(x))}{log(x)})^{\beta}$

So I got this function and I'm looking for an asymptotic expansion for different values of$\ \beta > 1 $ $\ f(x)=\left(1- \beta \frac{\log \left( \log(x) \right)}{\log(x)} \right)^{\beta}$ as $\ x ...
1
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2answers
31 views

Show that the series converges ($l^2$)

I know that $\sum_{k=1}^\infty|y_n|^2=S<\infty$. I also have that $\lambda >1$. I need to show that $$ \sum_{k=1}^\infty \left| \frac{y_1}{\lambda^k} + ...
2
votes
1answer
21 views

If $\sum_{k=0}^{r-1} c_k =0 $, and $a_n \to 0$, does $\sum_{n=0}^{\infty} \sum_{k=0}^{r-1} c_ka_{nr+k} $ converge?

This is a generalization of the alternating series convergence result and this: Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Here is my question: If ...
2
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1answer
77 views

Mistakes in $\lim_{a\to \infty}(a^2 - a) = - \frac{1}{6}$?

One can say, using Ramanujan summation or the zeta function regularization, that the sum $\sum_{k=1}^{\infty} k=- \frac{1}{12}$. Using this result I've gotten a very confusing and counterintuitive ...
-1
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2answers
38 views

The sum of three natural numbers are $111$, and the three numbers are in geometric progression.

Find all triples of natural numbers $(a,b,c)$ such that $a,b$ and $c$ are in geometric progression, and $a+b+c=111$. Any pointers?
1
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1answer
21 views

Identity for the product of sequences

Very easy question, How to express $\prod_{i=0}^n \prod_{j=0}^i a_i a_j $ as a function of just one index? Incidentally, where to find identities for product of sequences? There's a lot on ...
1
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1answer
120 views

Cauchy Sequences--is the floor function of a Cauchy sequence also a Cauchy sequence?

Okay so say you have some Cauchy sequence (a_n). And c_n=[[a_n]], where [[x]] refers to the greatest integer less than or equal to x. Is c_n also a Cauchy sequence? This is what I've got so far, ...
2
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2answers
58 views

Find the limit of this sequence

$y_0=k$ where $k$ is a constant. $x_{n+1}=30-\dfrac{y_n}{2}$ $y_{n+1}=30-\dfrac{x_{n+1}}{2}$ Prove that $(x_n, y_n)$ converges to $(20, 20)$ for all values of $k$. My attempt: I wrote a computer ...
1
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1answer
25 views

Prove that $T_n(x)={}_2F_1\left(-n,n;\tfrac 1 2; \tfrac{1}{2}(1-x)\right) $

Prove that, for Chebyshev polynomials of the first kind, \begin{align} T_n(x) & = \tfrac{n}{2} \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor}(-1)^k \frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} ...
0
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0answers
13 views

Can convolution be used to measure the difference between two sequences?

Say I have an infinite sequence $S_1$ and another finite sequence $S_2$. If I calculate $$ E = S_1 ∗ S_2 $$ does it somehow reflect whether $S_2$ appears somewhere in $S_1$? What if an approximate ...
8
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0answers
57 views

show this sequence inequality $x_{2^n}$

Define the sequence $\{x_{n}\}$ recursively by $x_{1}=1$ and $$\begin{cases} x_{2k+1}=x_{2k}\\ x_{2k}=x_{2k-1}+x_{k} \end{cases}$$ Prove that $$x_{2^n}>2^{\frac{n^2}{4}}$$ I have ...
2
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1answer
44 views

Proving the Fibonacci sum $\sum_{n=1}^{\infty}\left(\frac{F_{n+2}}{F_{n+1}}-\frac{F_{n+3}}{F_{n+2}}\right) = \frac{1}{\phi^2}$ and its friends

In this article, (eq.92) has, $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_{n+1}F_{n+2}} = \frac{1}{\phi^2}\tag1$$ and I wondered if this could be generalized to the tribonacci numbers. It seems it can ...
0
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0answers
17 views

Show that if $\sum_{k=1}^m c_k =0 $, $\sum_{n=0}^{\infty} \sum_{k=1}^m \frac{c_k}{nm+k} $ converges.

This is a generalization of this: Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Here is my solution. To show that if $\sum_{k=1}^m c_k =0 $, ...
3
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0answers
42 views

Reorder this series to change its sum [duplicate]

If in the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\cdots$ the order of the terms be altered, so that the ratio of the number of positive terms to the number of negative terms in the first $n$ ...
2
votes
2answers
22 views

Is the series $\sum_{n=1}^\infty n^r \exp(-k\sum_1^n \frac{1}{m})$ convergent when $k>r$ and $r<k$?

The series is: $$\sum_{n=1}^\infty n^r \exp(-k\sum_1^n \frac{1}{m})$$ The problem is asked to investigate this series when $r>k$ and $r<k$. However: $$ \frac{u_{n+1}}{u_n}\approx ...
4
votes
3answers
60 views

Determine the value of $ p $ for which the following infinite series converges and for which it diverges.

Determine the value of $ p $ for which the following infinite series converges and for which it diverges: $$ \sum_{n = 2}^{\infty} \frac{\sqrt{n + 2} - \sqrt{n - 2}}{n^{p}}. $$ I don’t know how to ...
1
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6answers
150 views

Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series?

Is there someone who can show me how do I evaluate this sum :$$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$$ Note : In wolfram alpha show this result and in the same time by ratio test ...
4
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1answer
44 views

Cardinality of a set of natural sequences

Let $a=(a_n)_{n\ge 1}$ a sequence such that for every $n\ge 1$ we have: a) $a_n \in\mathbb{N}$ b) $a_n\lt a_{n+1}$ c) Exists $\displaystyle\lim_{n\to \infty} \frac{\#\{j\mid a_j\le n\}}{n}$ Let ...
3
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3answers
85 views

Determine if this series $ \sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}$ converges

Determine if the following series converges: $$ \sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}. $$ (http://i.stack.imgur.com/qWiuy.png) I don't know how to start.
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2answers
46 views

Determine if $ \sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}$ congerges

Determine if the following series converges: $$ \sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}.$$ I'm supposed to use here the limit comparison test, but I don't know how to choose the second series.
3
votes
2answers
96 views

Some infinite series with Fibonacci numbers

An interesting problem is to prove that: $$ \sum_{n=1}^\infty \frac{F_{2n}}{n^2 \binom{2n}{n}}=\frac{4\pi^2}{25 \sqrt 5}. $$ I know the proof, which uses the fact that ...
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1answer
55 views

Prove the series converges a.s in Probability

I have an article as follows Why are they enough to prove that $ \sum_{n=1}^\infty \dfrac{X_n \textbf{1}_{\{|b_n|< |X_n|\}}}{b_n} $ converges almost surely? I want to know why must prove $ ...
4
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2answers
56 views

Choose initial values such that sequence always has integer values

We are given a recurrence relation defined by $$x_{n+2}=\frac{x_{n+1}x_n}{2x_n-x_{n+1}}.$$ Place necessary and sufficient values on $x_0$ and $x_1$ such that $x_n$ is an integer for all positive ...
1
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2answers
76 views

Sum of this Infinite Series

The series is as follows: $$\sum_{n=0}^\infty \frac{(-1)^nx^n}{(n!)^2}$$ I tried working on it. The square in the denominator is breaking me. Please If any one could help. And I need to find the ...
0
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1answer
46 views

Approximation for a binomial coefficient sequence summation

What is a good approximation to $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{\binom{k(k-1)/2}{i}}$$ $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{(\log ...
2
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3answers
106 views

Prove that a series is bounded with induction [duplicate]

I have to prove that the following condition is true: $$\frac{1}{n+1} + \frac{1}{n+2} + ... + \frac{1}{2n} > \frac{13}{24}$$ for every $n > 1$. I understood that this series is the same as: ...
0
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1answer
53 views

Pairs of integers with gcd equal to a given number

Given integers $N$ and $D$, find how many pairs of integers $(i, j)$ such that $1 \le i \le j \le N$ have the greatest common divisor exactly $D$. I know it involves Mobius inversion somehow, but I ...
3
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0answers
26 views

Show that ordered pairs are solutions to an equation if and only if they are consecutive elements of a recursive sequence (contest question)

The following question appeared on the 1998 Canada National Olympiad. I need help proving that the only solutions to the equation are consecutive elements of the recursively-defined sequence. I ...
2
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2answers
72 views

Is $\lim S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}<\infty $ for $ n \to \infty$ and $m$ large?

Let $m$ be a positive integer and let $$S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}$$ be a partial sum of real series . My question here is :Is $\lim S_{n,m} <\infty $ for $ n \to \infty$ and ...
3
votes
2answers
120 views

Why doesn't this work for Rudin Exercise 3.8

The problem is 3.8 exercise in baby Rudin: If $ \sum{a_n} $ converges and $\{b_n\}$ is bounded and monotonic, prove that $\sum{a_nb_n}$ converges. Why can't I just do this?: Let $M$ be an ...
3
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1answer
30 views

What is the closed form of the following expansion

I need some help figuring out the closed form of the following expansion. T[n]=T[n-1]+T[1]*T[n-2]+T[2]*T[n-3]+T[3]*T[n-4]+...+T[n-1] I haven't done this type of ...
8
votes
3answers
155 views

Sum of the series $\sum\limits_{n=0}^\infty \frac{1}{(3n+1)^3}$

The following result matches very good numerically: $$\sum_{n=0}^\infty \frac{1}{(3n+1)^3}=\frac{13}{27}\zeta(3)+\frac{2\pi^3}{81\sqrt{3}}.$$ Though I'm not sure how to approach this. How can we ...
0
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0answers
21 views

Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?

One of my friends asked me this question. "Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?" It is curious how even the most trivial questions ...
1
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1answer
38 views

Give an example of a filter that can not be generated by a sequence.

As the title I'm looking for an example of a filter that can't be generated by a sequence. If you took it from somewhere provide the source please. Expanding: Every sequence can generate a filter ...
0
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0answers
35 views

How to prove the convergence of a sequence satisfying $a_{k+1} \leq c_{1} a_{k} + \frac{c_{2}}{a_{k}} +1$?

Assume a positive sequence $\left\{ a_{k} \right\}$ satisfying \begin{equation} a_{k+1} \leq c_{1} a_{k} + \frac{c_{2}}{a_{k}} +1, k \in \mathbb{N} \end{equation} where $c_{1},c_{2},a_{1} > 0$. ...
0
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1answer
17 views

Repdigit sequences

Is there a formula to determine the probability of a sequence of repdigits in a longer sequence of random numbers? The Feynman point in $\pi$, for example, occurring within the first $1{,}000$ ...
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0answers
13 views

Problem in sequence [on hold]

Can anyone please tell me what to do when you are not able to solve a lot of like problems ,
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0answers
24 views

Expression for negating every other odd number index [on hold]

Is there a way to have an iterative expression that negates every other odd number index (starting from 3)? Basically, I am trying to write a generative expression that will give me value, given ...
5
votes
2answers
111 views

Wanted: more ways to prove $1+2x+3x^2+\cdots=(1-x)^{-2}$

If $|x|<1 $ prove that $\\1+2x+3x^2+4x^3+5x^4+...=\frac{1}{(1-x)^2}$ 1st proof:suppose ...
2
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0answers
55 views

How to integrate the following sum?

I'm currently trying to show: $$ \int_0^1{\int_0^y{\sum_{n=0}^{\infty}\left(\frac{1}{10^{n+1}x(1-x)}\left(9+\frac{1}{1-x^{10^n}}-\frac{10}{1-x^{10^{n+1}}}\right)\right)dx}dy}=\frac{10}{99}\log(10) $$ ...
3
votes
1answer
122 views

Limit of quotient of two infinite series $\left(\frac{0}{0}\right)$

Let $\sum_{n=1}^\infty a_{n,k}<\infty$. I want to calculate $$L=\lim_{k\to \infty}{\sum_{n=1}^\infty a_{n,k}\over\sum_{n=1}^\infty a_{n,k+1}}$$ if I know that $\lim_{k\to \infty} ...
2
votes
1answer
45 views

Lim sup/inf of average value

Consider $$f(t)= \frac{1}{t} \int_{0}^t \sin(e^s) ds.$$ What is $$\mathrm{lim \ inf}_{t \rightarrow \infty} f(t)$$ and $$\mathrm{lim \ sup}_{t \rightarrow \infty} f(t)?$$ Using $u$-substitution, ...
5
votes
1answer
84 views

How to evaluate this double infinite sum (Catalan number)

Let $C_n = \dfrac{1}{n+1}\binom{2n}{n}$. Is it possible to find the exact value of this infinite sum ? $$\sum_{n=1}^\infty \sum_{k=n}^\infty ...
1
vote
3answers
64 views

Finding convergence of a series using integral test

The series:$$\sum_{n=1}^{\infty}\left(\frac{\ln(n)}{n}\right)^{2}$$ Question: a) show that it converges b) find the upper bound for the error in approximation $s\approx s_{n}$ Trial: The section ...
1
vote
1answer
29 views

Asymptotic Expansion of $\ f(x)=\frac{\log(x)}{\frac{\log(x)}{2\alpha}-\log(\log(x))}$

I'm looking for the asymptotic expansion as $\ x \rightarrow \infty$ for $\ f(x)$ for small $\alpha$. Ideally, I'd like to get the asymptotic expansion for all orders. How would I go about doing this? ...
0
votes
1answer
14 views

Operator for comparing an n-tuple

Suppose you have to compare the following two finite ordered list of elements (tuples): $(\psi_{i}, R_{i}, A_{i}, \eta_{i})$ and $(\psi_{i}^{*}, R_{i}, A_{i}, \eta_{i})$ and for instance it turns out ...
2
votes
1answer
58 views

How to find the Summation S

Given function $f(x)=\frac{9x}{9x+3}$. Find S: $$ S=f\left(\frac{1}{2010}\right)+f\left(\frac{2}{2010}\right)+f\left(\frac{3}{2010}\right)+\ldots+f\left(\frac{2009}{2010}\right) $$
0
votes
1answer
24 views

Geometric Progression of Air removed by an Air Pump

If one third of the air in the tank is removed by each stroke of an air pump, what fractional part of the total air is removed in 6 strokes Answer is 0.0877 I was thinking this was some sort of ...
11
votes
5answers
982 views

Sum of an infinite series $(1 - \frac 12) + (\frac 12 - \frac 13) + \cdots$ - not geometric series?

I'm a bit confused as to this problem: Consider the infinite series: $$\left(1 - \frac 12\right) + \left(\frac 12 - \frac 13\right) + \left(\frac 13 - \frac 14\right) \cdots$$ a) Find the sum $S_n$ ...