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

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Asymptotic Estimates for a Strange Sequence

Let $a_0=1$. For each positive integer $i$, let $a_i=a_{i-1}+b_i$, where $b_i$ is the smallest element of the set $\{a_0,a_1,\ldots,a_{i-1}\}$ that is at least $i$. The sequence ...
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1answer
8 views

Absolute value of infinite series sum

How does it come to about that $|\Sigma_{n=-N}^{N}c_n(f)e^{inx} - \Sigma_{-\infty}^{+\infty} c_n(f)e^{inx}| = |\Sigma_{|n|>N} c_n(f)e^{inx}|$? What and why happens with the n-index? Could someone ...
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2answers
24 views

series comparison test

Is this correct? Q:Determine $\sum_1^n$$\frac{2}{3+5n}$ converges or diverges. A:$\frac{2}{6n}$ < $\frac{2}{3+5n}$ , since $\sum_1^n$$\frac{2}{6n}$ is a harmonic series and diverge, then ...
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1answer
62 views

Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$?

I hope this is not a duplicate question. If we modify the well known geometric series, with $a>1$, to $$ \sum_{n=0}^{\infty} \frac{1}{1+a^n} $$ is there a closed form with a name? I suspect ...
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0answers
23 views

Remainder Estimate for Integral test

I have the following question, it is a fill in the blank type question, however when I submit my answer, the system which verifies it say it is incorrect. I believe I am right, so I was hoping for ...
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0answers
16 views

A Sum into a Product, a polynomial into a trigonometric form

I've recently taken interest in infinite products, and I'm having trouble with a proof I found in this PDF file: "Infinite Products and Elementary Functions": An intermediate step in finding an ...
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1answer
59 views

Question about two sequences with a common limit

Suppose $a _n$ is a sequence of positive integers such that $ \lim_{n \rightarrow \infty} (a_n)^{\frac{1}{n}} $ exists. Suppose there exists a sequence of positive integers $ b_n $ such that $$ a_n = ...
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32 views

Additional explanation needed for the solution to a spesific sequence

Some of my attempts include trying to use the the term formula for geometric sequences and some other manipulations in hope of getting a more clearer, workable expression, though without success. ...
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3answers
82 views

Find the formula of the sum of $\frac{1}{1+x^2} + \frac{1}{(1+x^2)^2} + \dots + \frac{1}{(1+x^2)^n}$

How would I find the sum of this geometric series: $$ \frac{1}{1+x^2} + \frac{1}{(1+x^2)^2} + \dots + \frac{1}{(1+x^2)^n} $$ I want a formula, in the form of $\frac{n}{n+1}$, that can be proven by ...
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2answers
63 views

Find the maximum value of the fraction

Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b}<\frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime ...
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2answers
45 views

Proving by induction $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$

The context is as follows: I am asking this question because I would like feedback; I am a beginner to mathematical proofs. We wish to show $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + ...
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0answers
34 views

Why is it so hard to find a generating function for Somos' sequence?

The sequence is $\{1,2,12,576,1658880,\dots\}$. The $n$th number is obtained by squaring the $(n-1)$-th number and multiplying by $n$. So we start with $a_1=1$, $a_2=1^22=2$, $a_3=(1^22)^23=12$. In ...
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0answers
12 views

Partial sum of exponential series strictly increases after certain step (repost)

I apologize for the repost, previous formulation was not precise unfortunately. While trying to show that partial exponential series evaluated at two different ...
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3answers
36 views

Difficult nonlinear system based on max value

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 - xyz = 2$, $ y^3 - xyz = 6$, $ z^3 - xyz = 20$. The greatest possible value of $ a^3 + b^3 + c^3$ can be written in the form $ ...
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3answers
41 views

Does the following series converge uniformly?

I know how to show that the following series will converge absolutely. But am unsure how to show it will or will not converge uniformly for $z\in (0,1).$ $\displaystyle \sum_{n \mathop = 1}^{\infty} ...
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1answer
47 views

Infinite Telescoping Sum: $\sum_{i=1}^{\infty} (X_i - X_{i-1})=$?

Let $(X_i)_{i \geq 0}$ be any countable sequence of numbers and suppose that a limit exists, so $$\lim_{i \rightarrow \infty} X_i = x.$$ Consider $\sum_{i=1}^{\infty} (X_i - X_{i-1})$. Is this ...
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0answers
48 views

Counting the sum $\sum^{\infty}_{k=0} q^{k^{2}}$

Is it possible to obtain explicit form of the sum $\sum^{\infty}_{k=0} q^{k^{2}}$ (without using elliptic functions)? It is well known that $\sum^{\infty}_{k=0} q^{k} = \frac{1}{1-q}$ for all $q \in ...
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0answers
28 views

Why does this hold: $ \sum^{\infty}_{n=0}\frac{x^n}{n!}\mathbf e_{n}(y)+\sum^{\infty}_{n=1}\frac{y^n}{n!}\mathbf e_{n-1}(x) = e^{x+y} $

In the book "Stochastic Processes for Insurance and Finance" by Rolski et al. the following identity is used: $$ \sum^{\infty}_{n=0}\frac{x^n}{n!}\mathbf ...
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2answers
14 views

Series of positive factors of a number divided by that number

Let $S_n$ be the sum of the positive factors of $2015^n$, with $n$ being a positive integer approaching infinity. What is $\dfrac{S_n}{2015^n}$? I might be on the wrong track, but I figure that if $x ...
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0answers
29 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 $\sum_{n=0}^{\infty} (\frac x2)^n$, ...
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2answers
41 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 ...
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2answers
36 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
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1answer
23 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
81 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 ...
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2answers
44 views

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

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?
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1answer
23 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 ...
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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, ...
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2answers
61 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 ...
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1answer
26 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} ...
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0answers
17 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 ...
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+50

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 ...
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1answer
45 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 ...
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1answer
33 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 $, ...
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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 $n$ ...
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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
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3answers
61 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 ...
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6answers
152 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
48 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 ...
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3answers
87 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
47 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
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2answers
104 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
59 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
57 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 ...
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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
55 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
78 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
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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 ...