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

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29 views

Prove that all the five sequences converge to the same point $P \in \mathbb{R}^3$.

Let five sequences $A_n, B_n, C_n, D_n, E_n \in \mathbb{R}^3$ be constructed as follows: $A_0, B_0, C_0, D_0$ and $E_0$ are some given points of the space and $A_{n+1}, B_{n+1}, C_{n+1}, D_{n+1}, ...
2
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0answers
20 views

How many Fibonacci Numbers are in the sequence

I have $I_n = \{2^n + 1, 2^n + 2, 2^n + 3, \dots , 2^{n+1}\}$ and I am trying to prove using induction how many Fibonacci numbers are there. First, the length of $I_n$ is $|I_n| = 2^n$ then for $F_0 ...
3
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3answers
42 views

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$ How do I simplify this limit? I tried multiplying dividing $\sin(\frac{x}{2^r})$ to use half angle formula but it doesnt give ...
2
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1answer
43 views

Looking for formula of $\sum_{k=1}^m (-1)^k \dfrac {x^2(x^2-1)…(x^2-k+1)}{(x+1)(x+2)…(x+k)}$

Let \begin{equation*} u_k:=(e-1)^k \dfrac {x^2(x^2-1)...(x^2-k+1)}{(x+1)(x+2)...(x+k)}. \end{equation*} Can we find the sum of first $m$ of $u_k$ 's? That is, is there any formula for $\sum ...
0
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1answer
24 views

How to calculate the number of combinations of $x$ integers, each with a value between $y$ and $z$?

For example, if I have 4 integers, and each can be between 0 and 36, how many combinations are there? If the numbers have appeared before, but in a new order, then this still counts as a new ...
2
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2answers
31 views

Computing the value of a series by telescoping cancellations vs. infinite limit of partial sums

$$\sum_{m=5}^\infty \frac{3}{m^2+3m+2}$$ Given this problem my first approach was to take the limit of partial sums. To my surprise this didn't work. Many expletives later I realized it was a ...
4
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1answer
20 views

How to make a topology out of $N$ that involves convergent / divergent sets.

Let $N$ be the naturals $1, 2, \dots$ Call a subset $A$ of $N$ convergent if the reciprocal sum $\sum_{a \in A} \frac{1}{a}$ converges. Similarly call as set divergent if the sum diverges. Notice ...
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0answers
30 views

Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$

Let $(x_n)$ be a sequence in $X$ and $x \in X$. $(X,d)$ is a metric space and $f: X \times X\to $ is a bijection. Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$. ...
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1answer
32 views

About inequality $\sum_{k=1}^n |a_k|^2 \lessgtr \sum_{k\neq s} |a_k| |a_s|$

Let $a_k$ a sequence of complex number. We have $$\left(\sum_{k=1}^n |a_k|\right)^2 \geq \sum_{k=1}^n |a_k|^2$$ It is a basic fact because $$\left(\sum_{k=1}^n |a_k|\right)^2 = \sum_{k=1}^n |a_k|^2 + ...
5
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1answer
55 views

Cauchy-Ramanujan Formula $ \displaystyle \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} $

Cauchy and Ramanujan both gave the formula: $$ \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} = (2\pi)^{4p+3}\sum_{k=0}^{2p+2} (-1)^{k+1} ...
1
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3answers
44 views

Explicit form of this series expansion?

I am considering the following series expansion: $$f(k):=\sum_{n\geq 1} e^{-k n^2}$$ with $k>0$ a fixed parameter. Is there a possibility to either find a closed form expression for $f(k)$? Or at ...
2
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1answer
33 views

Find an example that the following equality doesn't apply

I need a sequence $(f_n)_{n\in\mathbb{N}}, f_n:X\to\mathbb{R}^\star$ which is measurable, $f_n \ngeq 0$ and doens't accept the following equality: $$\int_X\sum_{n=1}^\infty f_n \, d\mu = ...
1
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1answer
9 views

Sequence of functions that extends the algebraic properties of exponents to higher level operators.

I was thinking about some simple algebraic exponent properties such as the following $$ z^{x+y} = z^xz^y $$ and I started wondering about analytically continuing this identity to "higher-level ...
2
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0answers
35 views

Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
1
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4answers
106 views

The convergence of the series $\sum (-1)^n \frac{n}{n+1}$ and the value of its sum

This sum seems convergent, but how to find its precise value? $$\sum\limits_{n=1}^{\infty}{(-1)^{n+1} \frac{n}{n+1}} = \frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+...=-0.3068... $$ Any help ...
2
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1answer
23 views

Normal convergence: $\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$

I want to prove that: $$\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$$ is not normally convergent on $[a, \infty)$ for fixed $a>0$. Let $U_n(x)$ denote the general term. We have: ...
0
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1answer
49 views

Help with sequence $1\cdot n + 2(n-1) + \ldots + (n-1)2 + n\cdot 1$

Can anyone please provide a simplified formula for the sum of the sequence \begin{equation*} s(n) = 1\cdot n + 2(n-1) + \ldots + (n-1)2 + n\cdot1 \end{equation*} where $n$ is an integer greater than ...
1
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1answer
59 views

Difficult sequence puzzle

What is the 25$^{th}$ term of this infinite sequence? $$1,1,1,1,1,691,2,3617,...$$ I have tried for an hour now and I can't find any meaningful relation between the terms.
2
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2answers
52 views

How to evaluate $\sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty \frac{1}{j^2-k^2}$

I was reading an introductory text on multiple integrals and I have encountered a problem asking me to explain why $$ \sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty ...
2
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3answers
37 views

How to prove that $\sum \sin(2\pi n! x)$ does not converge uniformly on $\mathbb Q$

I want to prove that: $$\sum_{n \ge 1} \sin(2 \pi n! x)$$ converges absolutely on $\mathbb Q$, but not uniformly. For absolute convergence, let $x \in \mathbb Q$, then we can write: $x = a/b$ for ...
1
vote
1answer
35 views

Find new generating function, given an arbitrary generating function

In a discrete mathematics past paper, I am asked to find the generating function for the sequence $$\langle a_0, 0, a_2, 0, a_4, 0, \ldots \rangle,$$ given that $A(x)$ is the generating function for ...
0
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0answers
29 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
5
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2answers
53 views

Find the generating function of this sequence

I need to find the generating function of the sequence $c_n = (a_0, a_1, a_2, \ldots)$, where: $$a_n = \begin{cases} 2^{n/2} & \text{if $n$ is even,} \\ 1 & \text{if $n$ is odd.} ...
3
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1answer
46 views

If $\sum_n \frac{1}{\alpha_n}$ is convergent, what can we say about $\min_{r,s}{}_{+} |\alpha_r-\alpha_s|$?

Let $\alpha_n$ a sequence of real numbers. If $$\sum_n \frac{1}{\alpha_n}$$ is convergent, what can we say about $\min_{r,s}{}_{+} |\alpha_r-\alpha_s|$? I thought, for example, you could say ...
0
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1answer
17 views

Proof of Lagrange Identity

I need to prove Lagrange Identity for complex case, i.e. $$ \left( \sum_{i=1}^n|a_i|^2 \right)\left( \sum_{i=1}^n |b_i|^2 \right)-\left| \sum_{i=1}^na_ib_i \right|^2=\sum_{1\leq i<j\leq ...
5
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1answer
95 views

Does this sum converge or not?

I have the following sum:$$1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}-\frac{1}{9}-\frac{1}{10}+++++------...$$ How can I get an expression for its partial ...
2
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1answer
32 views

$\sum_{n=1}^\infty {a_n}$ is absolutely convergent and ${b_n}$ is any subsequence of ${a_n}$, then $\sum_{n=1}^\infty {b_n}$ is abs. convergent

If $\sum_{n=1}^\infty {a_n}$ is absolutely convergent and ${b_n}$ is any subsequence of ${a_n}$, then $\sum_{n=1}^\infty {b_n}$ is absolutely convergent. My attempt of proof: Let ${b_j}={a_{n_j}}$ ...
1
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2answers
89 views

$\sum_{n=1}^\infty {a_n}$ converges $\iff \sum_{n=1}^\infty {a_{n_k}}$ converges.

Let $({a_n})_{n\in{\mathbb{N}}}$ a sequence,and let $({a_{n_k}})_{k\in{\mathbb{N}}}$ the sequence of all terms of $({a_n})$ different than zero. Then $\sum_{n=1}^\infty {a_n}$ converges iff ...
0
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2answers
26 views

Alternating Reciprocal of Squares

I know that the infinite sum of the reciprocals of squares converges to $\pi^2/6$. Interested by this, I looked at a different sum. It is similar to the previously mentioned series, but it alternates ...
4
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1answer
32 views

Variant Generating Function related to Euler Numbers

The generating function $$\frac{2e^x}{e^{2x}+1}=\sum_{n\ge 0}E_k\frac{x^k}{k!}$$ counts the number of alternating permutations of a set with an even number of elements. My question is this, if we ...
3
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2answers
30 views

Triangular Array's Recursive Formula Breakdown

I have the following polynomials: $$1$$ $$z-1$$ $$z^2-2z+3$$ $$z^3-3z^2+9z-15$$ $$z^4-4z^3+18z^2-60z+93$$ $$z^5-5z^4+30z^3-150z^2+465z-725$$ $$...$$ They are generated both recursively and explicitly. ...
2
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3answers
52 views

Uniform convergence: $\sum \frac{x^2}{(1 + x^2)^n}$

Consider the series of functions: $$\sum_{n \ge 1} \frac{x^2}{(1 + x^2)^n}$$ Q: Where does this series converge uniformly? We have if $x \neq 0$: $$\lim_{n \to \infty} \left| \frac{x^2}{(1 + ...
4
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0answers
32 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
0
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1answer
16 views

Applicability of tests of convergence for series with non-negative terms

We know that there are many criteria of convergence for series with non-negative terms (for example, ratio test (with limit), root test (with limit), integral, comparison, and asymptotic comparison, ...
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0answers
44 views

$\sum_{n=0}^{\inf}n=- \frac{1}{12}$ Is it real? Why? [duplicate]

I assume many of you have stumbled upon the widely spread video that "proofs" the statement in my question. That video contains some serious mistakes (bracketing divergent series etc.), but the result ...
3
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2answers
33 views

Total area for a natural nested set of convex polygons.

Suppose we have a convex polygon $P_0$ with $n$ given vertices, and we want to "nest" polygons $P_j$ for $j > 0$ by taking the midpoints between edges of $P_{j-1}$ as the vertices. For a regular ...
2
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3answers
34 views

Determine a closed form for this sequence

Every year, 38 % of the amount of fish in a pond die. The 1st of May 2011 there were 5200 fish in the pond. Every year after May 1st 2011, 1900 new fish are added to the pond. Let $a_n$ be the amount ...
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0answers
27 views

help with showing a series is divergent [on hold]

I tried unsuccessfully to show by convergence tests that the series $$\sum_{n=1}^\infty{\ln^nn\over n^2}$$ is divergent , cant seem to find a way. help would be very appreciated , thanks in ...
2
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2answers
19 views

Proving two Recurrence Relations

I encountered this problem in the International baccalaureate Higher Level math book, and my teacher could not help. If anyone could please take the time to look at it, that would be great. I need ...
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1answer
59 views

How can I find out if a non-convergent series is “indeterminate” (that is, “oscillating”) or “divergent”?

Definitions: Given a sequence $\{a_n\}$, define $$s_n= \sum_{j=0}^n a_j.$$ The sequence $\{s_n\}$ is called the series of partial sums of $\{a_n\}$. A series is convergent if $\{s_n\}$ has ...
0
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1answer
28 views

Convergence of $\sum_{n=0}^\infty \frac{\sqrt{1+x^n}}{x^n}$ in the case $x<0$ and an analogous problem with $\sum_{n=0}^\infty \frac{x^n}{2+x^n}$

Let $$\sum_{n=0}^\infty \frac{\sqrt{1+x^n}}{x^n}.$$ My first question is: for what values of $x$ is this series possible? I can only say that it is not defined for $x = 0$, but are there other ...
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0answers
32 views

Help and solve for me on Sequences [on hold]

solve for me this sequence. The number of the sequence is 17, if the second number is 3, what is the first number in that sequence
0
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0answers
17 views

Algorithm for filling in points around a circle with increasing density

The aim of this question is to decide on the order in which to download a series of high-resolution files that together represent a 720° rotation around an animating object. When all the files are ...
-2
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0answers
32 views

$\textrm{lim sup}$ of a function [on hold]

I need to compute a limit: Given a series with the sum equal to $1$, I need to compute $\textrm{lim sup}\ (A_n)^{1/n}$. Since the sum is $1$, I assume that the value $A_n$ which actually should tend ...
0
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0answers
25 views

Algorithm for generating a series of contrasting colours

On a computer screen, colours can be defined as having 0-255 units of red, green and blue. This creates a 3-dimensional colour space with $256^3$ different colours, from 0-0-0 for black to 255-255-255 ...
1
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1answer
78 views

For which values of $q$ and $\alpha$ does $\sum_{n=0}^\infty q^{n^\alpha}$ converge?

Let us consider the series $$\sum_{n=0}^\infty q^{n^\alpha}, \qquad \qquad\qquad (*)$$ with $\alpha,\in \mathbb{Q}, n \in \mathbb{N}$. I know that the geometric series converges if $|q|<1$, but ...
2
votes
3answers
76 views

Does $\sum_{n=2}^\infty\frac{1}{n\ln n!}$ converge?

Let $$\sum_{n=2}^\infty \frac{1}{n\ln n!}.$$ It is equal to $$\sum_{n=2}^\infty \frac{1}{n(\ln n + \ln(n-1) +...+ ln(2))}.$$ But now what should I do to prove that it converges? (I have tried root ...
0
votes
1answer
25 views

How to show this is decreasing

I'd like to show $$\sum_{i=1}^n \frac{1}{i((n+1)-i)} $$ is decreasing for n>1, which is Cauchy product of $$\sum_{i=1}^n \frac{1}{i}$$ Numerical computation until n=50 shows it's decreasing but I ...
0
votes
1answer
32 views

Calculate the sum of the series $\sum_{n=0}^{\infty} \frac{1}{a_{n}\cdot a_{n+1}\cdot\ldots\cdot a_{n+7}}$, where $a_k = ak + b$

Let $a_k = ak + b$; define the following series: $$\sum_{n=0}^{\infty} \frac{1}{a_{n}\cdot a_{n+1}\cdot\ldots\cdot a_{n+7}}.$$ I have to prove that this series converges and I have to find its sum. ...
1
vote
2answers
28 views

Series definied by recurrence relation

Define a sequence recursively by $$\large{a_1 = \sqrt{2},a_{n+1} = \sqrt{2 + a_n}}$$ (a) By induction or otherwise, show that the sequence is increasing and bounded above by 3. Apply the monotonic ...