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

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21 views

Why zeta(2) in these inifinite sums?

The infinite sum of the reciprocals of these two sequences have zeta(2) in the result. The value is not in OEIS. A000326 A002411 $$\sum _{m=1}^{\infty } \frac{1}{\sum _{n=1}^m \frac{1}{2} n (3 n-1)}= ...
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3answers
41 views

Does $\sum_{n=0}^\infty (-1)^n (e-(1+\frac{1}{n})^n)$ converge absolutely, conditionally, or diverge?

I've been given the hint to use the binomial theorem and show that $e-\left(1+\frac{1}{n}\right)^n > \frac{1}{2n}$ for $n \geq 2$. So I've written \begin{align*} e-\left(1+\frac{1}{n}\right)^n = e ...
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2answers
31 views

tricky telescopic sum

Consider the sum of $\frac{1}{n+a-b}-\frac{1}{n+a}$ from $n=1$ to infinity. Show that is in fact an finite sum. I have written down some terms but can't see where cancellation is occurring.
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1answer
16 views

Product of bounded and convergent to $0$ sequence is a convergent to $0$ sequence

Let $a_n$ be a null sequence and let $b_n$ be a bounded sequence. Prove that $a_n \cdot b_n$ is a null sequence. I tried using the product rule of sequences but cannot because $b_n$ is not ...
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0answers
16 views

Series and sequences significance [on hold]

What is the significance of series and sequences like trigonometry series in nature
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0answers
22 views

Question about simple limit. [duplicate]

suppose $a_n>0, \quad \alpha \in (0,1),\quad k>0,$ and $\qquad \lim\limits_{n \to \infty} n^{\alpha}(\cfrac{a_n}{a_{n+1}}-1)=\lambda, \qquad \lambda\in(0,+\infty)$, prove $\lim\limits_{n \to ...
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0answers
31 views

The Riemann Zeta Function Works [on hold]

1 - Any counterexamples known for the Riemann Zeta Function? 2 - How to generalize the following? Here we have the visualization of the Riemann Zeta Function 3D Plot and the plane. We can observe ...
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1answer
28 views

Sequences of polynomial functions converging uniformly on $[a,b]$ to a continuous function not a polynomial

What is (are) the necessary and sufficient condition(s), if any, for a sequence of polynomial functions to converge uniformly on a given (finite) closed interval $[a,b]$ to a continuous function not a ...
3
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1answer
55 views

Proving that $\lim\limits_{n \to \infty} \frac{E_{n+1}}{E_n}=2^{-2/3}$

$$\def\ut#1{\underline{\text{#1}}}\def\vec#1{\mathbf{#1}} \def \d{\mathrm{d}} \def \p{\partial } \def \[{\left[} \def \]{\right]} \def \({\left(} \def \){\right)} \def \n{\boldsymbol{ \nabla}} ...
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3answers
62 views

How to evaluate this $1/n$ infinite sum?

How to evaluate$$\sum ^{\infty}_{n=1} {e}^{-n}$$ without using the easy-formula. We easily notice a pattern. $$\begin{align} S_1 &= e^{-1} \\ S_2 &= e^{-2} + e^{-1} = \frac{1 + e}{e^2} \\ ...
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2answers
24 views

terms asymptotically equal?

I need to prove that $\left ( \frac{k-i+1}{k}\right )^{j}$ and $\left ( \frac{k-i}{k}\right )^{j}$ are asymptotically equal when k is large enough, $1\le i\le Q$ and $Q$ is a constant. Could you ...
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2answers
33 views

Is this rigorous notation?

$$\sum_{n=-\infty}^{+\infty} f(x,n)$$ Is it rigorous to write this ? It feels weird to write "$n = -\infty$" ...
0
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1answer
13 views

mean and median of an arithmetic progression

Is the mean always equal to the median of an arithmetic progression e.g. for a set of consecutive integers x, x+1, x+2........,y-2, y-1, y The median is (x+y)/2 equals the mean (x + x+1 +.....+ y-1 ...
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2answers
16 views

If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3$ [on hold]

True or False: If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3$ Then: a) $\sum_{n=0}^\infty c_n 2^n$ converges. b) $\sum_{n=0}^\infty c_n 3^n$ converges. Can someone please give me any ...
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2answers
30 views

Prove $(x_n)$ does not converge to real number $x$ using definition

$x_n$ = $(-1)^n(1-{1\over n})$ Prove: If $x$ ∈ $\Bbb{R}$ is any real number, then ($x_n$) does not converge to $x$. This has to be proved using the negation of the definition i.e. ...
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1answer
34 views

Inequality depending on a series and a function

I know that $ \forall n\in N, a_0 + a_1 + ... + a_n = 1$, with $a_0, a_1, ... a_n > 0$ and $f(t) = a_0t^n+a_1t^{n-1}+...+a_n, \forall t\in R$ I have to prove that $$ f\left(\sqrt{x^3} \right)* ...
2
votes
2answers
74 views

Strenuous, arduous, laborious, onerous sum

$$\sum \limits_{n=1}^\infty \left(\frac1n{\sum \limits_{k=1}^n \frac1k}\right)^3$$ I want to see what this converges to. Obviously it converges on inspection. Any suggestions on methods to use?
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2answers
23 views

Prove (xn) does not converge to real number x [on hold]

$x_n$ = $(-1)^n(1-{1\over n})$ Prove: If $x$ ∈ $\Bbb{R}$ is any real number, then ($x_n$) does not converge to $x$. This has to be proved using the negation of the definition i.e. ...
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2answers
30 views

Finding limit of sequence

I have to find the limit of sequence ${a_{n}}$ such that $${a_{n}} = \frac {n^\frac{2}{3} \sin(e^n)} {n+1}$$ I have no idea where to start. Any hints on where to start?
2
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3answers
35 views

Inverse Matrices and Infinite Series

Given that $C=I+A+A^2+A^3+ \ldots$ Prove that I-A is the inverse of $C$ Hint: Use the infinite series technique for finding inverse of a matrix. Now I know with an infinite geometric series with a ...
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0answers
19 views

What is a “hypergeometric series” with differences, not just sums, of indices?

"Hypergeometric series" often have forms like (in two variables) $$\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{(a_1)_n (a_2)_n}{(b_1)_k (c_1)_{n+k}} \frac{x^n}{n!} \frac{y^k}{k!}$$ And there are ...
2
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0answers
20 views

Function from a limit of a trigonometric sequence

Let $0\le x\le\pi/2$ and let $a,b\in (0,1]$ be real parameters. Define the sequence $$ \theta_0 = x \\ \theta_{n+1} = \arcsin\left(a\sin(b \theta_n)\right) $$ for $n\ge 0$. If $a=b=1$ the sequence is ...
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0answers
34 views

Harmonic number identity

I search for an elementary proof of the following identity: $$ \sum_{i=1}^{n-k} \frac{(-1)^{i+1}}{i}\binom{n}{i+k}=\binom{n}{k}\left(H_n-H_k\right) $$ I have found the following proof: $$ ...
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1answer
25 views

Sets of binary sequences

In my course on linear algebra we have recently introduced linear independent subsets of vector spaces. As an exercise I have been thinking about examples of infinite linearly independent sets and ...
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0answers
34 views

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
9 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 ...
3
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1answer
55 views

Why do the averages of $\sin (p+\cos p )$ approach a positive limit?

$$S_1 = \sum_{k=1}^n (\sin \left[~ p(k) + \cos p(k)~ \right])$$ I wonder why this appears to give $$\frac{1}{n}S_2\sim 1/2 $$ Thanks for any insights or references. Edited in light of ...
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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 ...
2
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2answers
35 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
30 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
47 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 ...
2
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2answers
29 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 ...
0
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1answer
18 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 ...
0
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1answer
36 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 ...
2
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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
35 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
64 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 ...
2
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4answers
70 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 ...
2
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3answers
56 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
41 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 ...
0
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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
43 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
30 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
32 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 $M\in N$, ...
0
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1answer
33 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
19 views

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
40 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
votes
1answer
40 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 ...
0
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2answers
25 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
votes
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 ...