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

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Are there ever cases where it's easy to get coefficients for the series representation for an integrand, but hard to approximate the integral?

WHY I'M ASKING THIS I'm working on a faster way to approximate integrals of series. So I'd like to know if this could be useful. THE QUESTION If we suppose that we can get a formula for the ...
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1answer
5 views

Boundedness on k-tuple euclidean space

I am currently studying, "Elementary Analysis:The Theory of Calculus" by Kenneth A. Ross, in my edition on page 82, bounded sequences in $\Re^k$, k-tuple Euclidean space is defined as follow: A set S ...
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0answers
15 views

Why do these two sums behave differently?

My computer treats the following two sums very differently and if someone could give me a brief idea of why this is the case I'd be interested. $p(k)$ is the kth prime. $$S_1 =\sum_{k=1}^n (\sin k + ...
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1answer
12 views

Close form of a summation of a sequence to infinity

I am trying to find the closed form of the following summation \begin{equation} \sum_{i=0}^{\infty}(-a)^i\frac{\Gamma(M+i)}{\Gamma(N+i)i!} \end{equation} where $a$ is a real number, $\Gamma(\cdot)$ is ...
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2answers
33 views

Does $\sum\limits_{k=1}^{\infty} (a_{k+1} - a_k)^2 <\infty$ imply $\frac{1}{n^2} \sum\limits_{k=1}^n a_k^2 \to 0$?

I'm currently working on some problem regarding Dirichlet forms and, as the title states, I'm trying to figure out if $$\sum\limits_{k=1}^{\infty} (a_{k+1} - a_k)^2 <\infty \Rightarrow ...
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0answers
26 views

Taylor Series Expansion for Function of Two Variables (with Countable Discontinuities)

Given a real-valued function of two real variables, under certain conditions of smoothness in a closed ball about some point, we can obtain a Taylor series for the function about that point. I want ...
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2answers
43 views

Is this sequence decreasing?

If a sequence $b_n>0$ and $b_n$ converges to $0$, can we say it is eventually decreasing? This problem bumps up when I am trying to something bigger. However, I am very unsure of this. If this is ...
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2answers
24 views

How to prove or disprove this infinite sum of Bessel functions is zero

The sum is $$\sum_{n>0} \mathrm{i}^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$ and $$\sum_{n>0} (\mathrm{-i})^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$ I suspect they ...
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1answer
17 views

How to Solve this Arithmetic Progression Question?

Please help- Four different integers form an increasing AP.One of these numbers is equal to the sum of the squares of the other three numbers.Then- find all the four numbers. I assumed the numbers ...
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1answer
32 views

Find a specific series with a known series

Let $\sum a_n$ be a convergent, positive series. Show that there exists a a convergent, positive series $\sum b_n$ such that $$\lim_{n\to\infty}\frac{a_n}{b_n}=0.$$ Do ...
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1answer
22 views

Help clearing up the definition of Limsup?

I was thinking about the equivalence of the two following definition of Limsup of a sequence. I find the definition 1 much more intuitive and I have been trying to convince myself of the equivalence ...
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2answers
34 views

trouble solving this sequence problem

I'm having some trouble solving this problem about sequences: a(n): a(1) = 2; a(n+1) = (a(n) + 1)/2, n belongs to N(natural numbers) 1)Prove that this sequence is monotonically decreasing 2)Prove ...
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3answers
60 views

The sum of all the odd numbers to infinity [duplicate]

We have this sequence: S1: 1+2+3+4+5+6.. (to infinity) It has been demonstrated, that S1 = -1/12. Now, what happens if i multiply by a factor of 2? S2: 2+4+6+8+10+12.... (to infinity). I have ...
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4answers
58 views

Find the sum of an infinite series of Fibonacci numbers divided by doubling numbers. [duplicate]

How would I find the sum of an infinite number of fractions, where there are Fibonacci numbers as the numerators (increasing by one term each time) and numbers (starting at one) which double each time ...
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3answers
55 views

For the series $S = 1+ \frac{1}{(1+3)}(1+2)^2+\frac{1}{(1+3+5)}(1+2+3)^2$…

Problem : For the series $$S = 1+ \frac{1}{(1+3)}(1+2)^2+\frac{1}{(1+3+5)}(1+2+3)^2+\frac{1}{(1+3+5+7)}(1+2+3+4)^2+\cdots $$ Find the nth term of the series. We know that nth can term of the ...
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2answers
39 views

How to properly state as to why $\sum_{n=1}^{\infty}\frac{\sqrt{n^3+2}}{n^4+3n^2+1}$ converges.

So I know that $\sum_{n=1}^{\infty}\frac{\sqrt{n^3+2}}{n^4+3n^2+1}$ converges, because the highest power in the numerator is $n^\frac{3}{2}$ and the highest power in the numerator is $n^4$, so I have ...
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1answer
12 views

Cauchy sequence of vectors when dotted with another vector gives a Cauchy sequence of scalars?

My question is related to vector spaces with an inner product defined (the space is not necessarily complete i.e. not a Hilbert Space) So imagine I have a Cauchy sequence of vectors ...
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2answers
42 views

How to determine whether $\sum_{n=1}^{\infty}\ln\left(\frac{n+2}{n+1}\right)$ converges or diverges.

I am trying to find whether $\sum_{n=1}^{\infty}\ln\left(\frac{n+2}{n+1}\right)$ converges or diverges. I used the limit test, and it comes out as inconclusive since ...
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0answers
27 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...
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1answer
31 views

$\{p_{n}\}$ is a sequence of real numbers. Prove $\limsup$ $\{p_{n}\} < \infty$ if and only if $\{p_{n}\}$ is bounded above.

I have done the following. $\Leftarrow$ $\limsup$ $\{p_{n}\}$ is the set of suprema of all the subsequential limit points of $\{p_{n}\}$. So, if it were not finite, then, given any $M\in N$, ...
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1answer
32 views

Calculus sequences And series

Find the values of $x$ for which the series $\sum_o^\infty \frac {(x+3)^n}{2^n}$ converges. I took it as $(\frac {x+3}2)^n$ then used the rule of summation of $r^n= \frac 1{1-r}$ then found ...
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0answers
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Name for hypergeometric-like sum.

Consider the sum $F(a_1,\cdots,a_r; z)\sum_{m=0}^{\infty}\frac{(a_1+m)(a_2+m)\cdots(a_r+m)}{(a_1)(a_2)\cdots (a_r)}\frac{z^m}{m!},$ where $a_i$ for $i=1,2,\cdots,r$ is an increasing sequence of ...
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0answers
39 views

Can this convergent series be generalised?

A friend of mine gave this question,I have no idea how to even start generalising the nth term of the series so that I can summify it to n tending to infinity. $$\frac{1}{(1!)} ...
2
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1answer
35 views

Proving these series equations to be equal?

I was recently attempting to prove the formulae which calculate the sum of arithmetic sequences where the difference between each term is just 1. I arrived at this formula first, which calculates the ...
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2answers
24 views

Prove that a sequence of recursive functions $\,f_n(x)$ cannot converge pointwise to $\,f(x)$ on $[0,1]$

Given a recursive sequence $\,f_n(x) :[0,1] \to \mathbb R$, $x \in [0,1]$, where $$\begin{align*} f_1(x) &= x, \\[6pt] f_n(x) &= \frac{2x\,f_{n-1}(x)}{n!} \end{align*}$$ I have proven that the ...
2
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3answers
59 views

Convergence of $\frac{\ln(n)}{n^2}$

During a Calc exam, I needed to state whether $$\sum_{n=1}^{∞} \frac{\ln n}{n^2}$$ converged or diverged. I was going to try a strict comparison test (with $\sum_{n=3}^{∞} \frac{\ln n}{n^2}$ to ...
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1answer
17 views

Limit of sequence involving a product

This question is related to a post that was deleted. I want to calculate the following limit $$\lim_{n \to ...
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4answers
48 views

An infinite series question. [on hold]

Ok, so we have an infinite sequence: S1 = 6+14+22+30+38.. Now, there is another infinite sequence, S2 = 1+2+3+4+5+6+7.. We know that the first sequence's nth term = (8n-2 ) so surely there must be a ...
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0answers
19 views

Rearranging a sequence to minimize a series function on its subsequences

Given an arbitrary nonempty sequence of integers $Q$ with $p$ elements: Let $R$ be a rearrangement of $Q$. Let $A$ be the solution set of $1\leq n\lt m\leq p$, where $n,m\in \mathbb{Z}$. $F(n,m) = ...
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1answer
18 views

How to prove that f(x)=1/|x-t| is continuous but not bounded?

Suppose S is not closed: there is a point t in R, t not in S, such that a sequence in S converges to t. Show that the function f: S-> R, defined by f(x) = 1/|x -t|, is continuous but not bounded.
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0answers
22 views

Showing a series is not the fourier series of a riemann integrable function.

I want to show that the series $\sum_1^\infty \frac{sin(nx)}{\sqrt{n}}$ is not the Fourier series of a Riemann integrable function on $[-\pi,\pi]$. I was going to do this by showing that the partial ...
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6answers
70 views

Prove that the sequenze $b_n=\left(1+\frac{1}{n}\right)^{n+1}$ is decreasing [duplicate]

Prove that the sequence $$b_n=\left(1+\frac{1}{n}\right)^{n+1}$$ Is decreasing. I have calculated $b_n/b_{n-1}$ but it is obtain: $$\left(1-\frac{1}{n^2}\right)^n \left(1+\frac{1}{n}\right)^n$$ But I ...
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3answers
41 views

Convergence series $\sum_{n=1}^\infty n\sin\left(\frac{1}{n^2}\right)$ [on hold]

You can help me to show if the following series converges or diverge. $$\sum_{n=1}^\infty n\sin\left(\frac{1}{n^2}\right)$$
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2answers
28 views

Real convergent sequences

Let $(a_n)$ be a bounded sequence for all $n$ such that $ \displaystyle a_n \geq \frac{1}{2} (a_{n-1}+a_{n+1})$ for $n\geq 2$. Show that $(a_n)$ converges. I think I cannot use any convergence tests ...
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0answers
16 views

Paperfolding Constant

In the Wikipedia article about the regular paperfolding sequence, it says (more or less quoted): Taking $G(t_n;x) = G(t_n;x^2) + \sum_{n=0}^{\infty}x^{4n+1} = > G(t_n;x^2) + \frac{x}{1-x^4}$ ...
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1answer
38 views

prove if is converges or diverges sequence

find if it's converges or diverges, if converges find the limit: $$\frac{(-1)^n n+1}{n^2+1}.$$ My proof: divided by n^2 so you have $\frac{(-1)^n(1/n+1/n^2)}{1+1/n^2}$ if I take the limit $n\to ...
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1answer
87 views

L'Hopital's Rule, Factorials, and Derivatives

I have the following limit $\displaystyle\lim_{n\to\infty}\frac{e^n}{n!}$. Now if I try to solve this using this using L'hopital's rule, I won't be able to since I can't take the derivative of $n!$. ...
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1answer
17 views

Calculate duration of task

Say I have some task to process 100 days of data, and it takes 5 hrs to process a day. But each day that it takes to process it a new day of data comes in. So for the initial set of data it takes: 5 ...
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1answer
21 views

Upper and Lower Limits of a Sequence

If we partition a sequence into a finite number of subsequences then the upper and lower limit of the sequence are equal to the maximum upper limit and minimum lower limit of the subsequences. Has ...
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0answers
17 views

Any suggestions to find a non recursive formula for this sequence?

I have these elements in a set $R$, $$\{r_i\ /\ i=1,2,3,\dots,2n \}\in R$$ And we define: $$a(1)=r_1+r_2$$ $$a(2)=\frac{(r_1+r_2)\cdot r_3}{(r_1+r_2)+ r_3}+r_4$$ ...
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1answer
15 views

Summation of arithmetic-geometric series of higher order

There is a closed formula for the summation of arithmetic-geometric series: $\sum_{x=1}^{+\infty }(ax+b)r^x=\frac{(a+b)r-br^2}{(1-r)^2}$ when $-1<r<1$ But consider: $\sum_{x=1}^{+\infty ...
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0answers
12 views

Expressing one generating function like combination of another generating functions.

Let A (t), B (t) and C (t) - generating functions for sequences $a_0, a_1, a_2,\dots; b_0, b_1, b_2,\dots and\ c_0, c_1, c_2,\dots$ Express C (t) through A (t) and B (t), if ...
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1answer
53 views

How to prove this sequence is convergent?

Suppose that the series $\sum_{k=1}^\infty{a_k}$ converges. Prove that $$\lim_{n→\infty}\frac{1}{n}\sum_{k=1}^{n}ka_k=0$$ I tried to use the definition of convergence of $\sum a_k$ ...
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0answers
26 views

If $a_i$ is an increasing sequence then which of the following is convergent?

If $a_i$ is an increasing sequence then which of the following is convergent? $\sum_{i=1}^{n}a_i$ $\sum_{i=1}^{n}\frac{a_{i+1}}{a_i}$ $\sum_{i=1}^{n}\frac{1}{a_i^2}$ $\sum_{i=1}^{n}\sqrt{a_i}$ I ...
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3answers
37 views

Calculating a limit with infinitely many terms

I've encountered this limit : $$\lim_{n\to\infty} \frac1n \left(\sin\left(\frac{\pi}{n}\right) + \sin\left(\frac{2\pi}{n}\right)+\cdots+\sin{\frac{(n-1)\pi}{n}}\right)$$ Wolfram gives the value: ...
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1answer
40 views

Arctan(f(x)) is almost the same as Erf(f(x)) for many f(x). Is the just coincidence or is there a reason?

For example: Arctan(x) is almost Erf(x) (subtle differences in absolute value and curve) Arctan(x^50) is almost Erf(x^50) (difference in absolute value) and many others, so we can conclude: ...
0
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1answer
10 views

Find the convergence radius for $ \sum_{k=0}^{\infty} \frac{k(z+i)^{2k}}{2^k} $

What's the convergence radius for $$ \sum_{k=0}^{\infty} \frac{k(z+i)^{2k}}{2^k} $$ I using the root criterium that says that the serie convergence if the limit is 0. $$ lim_{k \rightarrow \infty} ...
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2answers
32 views

convergence of $\sum_n x_n$ versus $\sum_n f(x_n)$ for differentiable $f$

Suppose $f(0) = 0$, $f$ is differentiable at 0, $f'(0) \ne 0$, and $x_n \rightarrow 0$. What can we say about (i) the convergence of $\sum_n x_n$ versus (ii) the convergence of $\sum_n f(x_n)$? It ...
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2answers
31 views

How to prove that $\sum_{i=0}^h2^i=2^{h+1}-1$ [on hold]

How do I prove the following relationship? $$\sum_{i=0}^h2^i=2^{h+1}-1$$
2
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1answer
19 views

Which of the following are true about sequences?

If $(x_n)$ is a sequence of real numbers such that for every $n$ we have $0<x_n<\frac{1}{n}$ then which of the following is true? $1.\lim_{n\to\infty}x_n=0$ $2.$If $f$ is continuous function ...