For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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2
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289 views

Prove that sum is finite with the help of generating function

Please help me to prove that the following sum is finite $$ \sum_{j=2l-2}^{\infty}j!\, a_j^{(l)}, $$ here the generating function of $\displaystyle{a_j^{(l)}}$ is ...
2
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131 views

Do we have closed form for these series?

Continuous on the previous question, can we get a closed form for these? $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( ...
2
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40 views

Compute $ \operatorname{Li}_{3}\left(\frac{1}{2} \right) $

Where could I find a proof of the identity $$ \operatorname{Li}_{3}\left(\frac{1}{2} \right) = \sum_{n=1}^{\infty}\frac{1}{2^n n^3}= \frac{1}{24} \left( 21\zeta(3)+4\ln^3 (2)-2\pi^2 \ln2\right)$$ ?
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333 views

Subseries of harmonic series

It is well known that harmonic series $$\sum_{n=1}^{\infty}\frac{1}{n}$$ diverges but in 1985 G. H. Behforooz proved that if we remove terms that have denominator that ends with $9$ series converges. ...
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80 views

Property similar to decreasingness

Let $\{ a_j \}_{j \in \mathbb{N}}$ be any sequence of real numbers satisfying the following property: There exists a number $N \in \mathbb{N}$ such that $a_k<a_j$ for all $k\geq j+N$. What ...
2
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277 views

Is Exercise 8.3 in Rudin's Principles of Analysis as easy as it seems?

Rudin Theorem 8.3 says that if $$\sum_{j=1}^\infty |a_{ij}| = b_i$$ and $\sum b_i$ converges, then $$\sum_i \sum_j a_{ij} = \sum_j \sum_i a_{ij}$$ Rudin 8.3 asks us to show that if $a_{ij} \geq 0$ ...
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177 views

Proof using Darboux's Definition: $\int_0^1f=\lim_{n\to \infty}\frac1n\sum_{i=1}^{n}f\left(\frac in\right)$

Let $f:[0,1]\to \mathbb{R}$ be Darboux integrable. I ask for a proof of $$\int_0^1f=\lim_{n\to \infty}\frac1n\sum_{i=1}^{n}f\left(\frac in\right)$$ where the integral in the left hand side is the ...
2
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193 views

Problem with understanding first (and second) derivative of a two-sided infinite series

For the function $$f(x)=b^x-1 = x_1 \qquad g(x)=\log(1+x)/\log(b) $$ and its iterative notation $$ x_0=x \qquad x_h=f(x_{h-1})=g(x_{h+1}) \qquad x_{-1}=g(x_0) $$ with b from the interval $1 \lt b \lt ...
2
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420 views

problem of uniform convergence of series

The problem is Prove that the series $$\sum_{n=1}^\infty (-1)^n\frac{x^2+n}{n^2}$$ converges uniformly in every bounded interval, but does not converge absolutely for any value of $x$. ...
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92 views

Identify the smallest $c$ such that $P(|X_n| \ge c \sqrt{\ln n} \text{ i.o.}) = 0$ for normally distributed $X_n$

The problem is to show that $P(|X_n| \ge c \sqrt{\ln n}\text{ i.o.}) = 0$ for standard normal $X_n$ that are not necessarily independent. Also, identify the smallest such $c$. I am thinking that the ...
2
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46 views

Formulating Cauchy Completeness in the category of linear spaces using categorical limits and colimits

Formulating Cauchy Completeness in the category of linear spaces using categorical limits and colimits, can it be done? I've made some naive attempts replace cauchy sequence with sum and taking the ...
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62 views

Convergence proof-how to apply this theorem?

I want to show that $\sum_{n=1}^{\infty}\frac{(-1)^n}{n+1}$ converges by using that theorem. Theorem: suppose (a) the partial sum $A_n$ of $\sum a_n$ form a bounded sequence (b) $\dots \le b_2 \le ...
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145 views

uniformly convergent

Define functions $f_n\colon [0,1]\to\Bbb R$ by $f_n(x)=n^px\exp(-n^qx)$ where $p$, $q>0$ and $f_n\to 0$ pointwise on $[0,1]$ as $n\to \infty$ and $\sup|f_n(x)|=(n^{p-q})/e$. Assume that ...
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199 views

Closed form of the series ,$\sum_{k=0}^{\infty}\frac{(-1)^k k!}{(k+1)^{(k+1)}}x^k$

$x,y>0$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{xt+e^{y t}} dt$$ if $x=0$ then $f(0,y)=1/y$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{e^{y t}(1+xte^{-y t})} dt=\int_{0}^{\infty} \frac{e^{-y ...
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66 views

tangents of infinite sums — reference request

This section of Wikipedia's List of trigonometric identities was, as far as I know, written entirely by me. For a time, it dealt only with finite sums, and said something like this: $$ ...
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377 views

Prove metric space…

Let $l=\{(a_n)_{n \in ℕ}|a_n \in \Bbb R \text{ and } \sup|a_n|<\infty\}$ If $a=(a_n)$ and $b=(b_n)$ are in $l$, define a metric on $l$ by $$d(a,b)=\sup_{n \in \Bbb N}|a_n-b_n|$$ Prove that $d$ is ...
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282 views

Number sequence - differences don't converge

I'm having problems with a particular number sequence: 175, 94, 188, 124, 376, 327, 1308, ... I've tried working out the differences and have came up with this ...
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77 views

how to show that a series converges without use of limits.

Just wanted to know if there is another method? Of the methods I know: Ratio test Comparison test Root test Integral test Limit comparison test All make use of limits. The reason why I am ...
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242 views

Calculate the sum of the first N terms of the sequence

$a_n=a_{n-1}\displaystyle \frac{n+1}{n}$ if $n > 1$ $a_n=1$ if $n=1$ I'm not too sure where to start here. This is part of a review for a class and I can't really seem to remember what we're ...
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60 views

Is exponent of discrete-analytic function also discrete-analytic?

Lets define a discrete analytic function such a function that is equal to its Newton series: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$ Is function $g(x)=e^{f(x)}$ also ...
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146 views

Convergence of a function in a metric space to its metric

Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
2
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118 views

sum of periodic function which eventually vanish

It is known that : If $\lim_{x \to \infty} \sum_{i=1}^n f_i(x)=0$ and $f_i$ are real-valued periodic function, then $\sum_{i=1}^n f_i(x)=0$ for all $x \in \mathbb{R}$. We can solve this problem by ...
2
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0answers
497 views

Does this ratio converge to the Golden Ratio?

This infinite sequence a(n) starting: $1, 1, 2, 2, 3, 4, 5, 7, 10, 14, 20, 30, 45, 68, 104, 161...$ is the antidiagonal sums of a triangle that has several properties in common with the Pascal ...
2
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0answers
64 views

Convergence of integral to convergence a.e question

OK, so now I have $$\int_0^T{\phi(t)h_n(t)} \to \int_0^T{\phi(t)h(t)}$$ which holds for all $\phi \in C^\infty_c(0,T).$ From this how can I deduce that $h_n \to h$ (a.e)? I think I need maybe ...
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144 views

Iterated Root Mean Square-Arithmetic Mean

Can I find iterated Root Mean Square-Arithmetic Mean as a function of Arithmetic-geometric mean (AGM) with some transformations if it is possible? if not possible, what is the closed form of it as ...
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358 views

Doubt $\sin(2n)/(1+\cos^4(n))$

So my doubt is for this comment of this video: Is ANYONE going to notice the fact that the kid's solution to the problem given at 9:35 is completely invalid? I mean, $\sin(2n)/(1+\cos^4(n))$ ...
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0answers
55 views

Why isn't the lag n autocovariance of an AR process infinity?

When calculating the lag 1 autocovariance of a simple AR process, if I define my $X_{t}$'s in terms of $ε_t$'s, I get something that looks a lot like an MA process with an infinite sum of standard ...
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0answers
113 views

A few questions about a relationship between some integer sequences and infinite recursive trees

In his book Gödel, Escher, Bach Douglas Hofstadter defines the following two integer sequences: Hofstadter G-sequence: $a(n)=n-a(a(n-1))$ Hofstadter H-sequence: $a(n)=n-a(a(a(n-1)))$ He says ...
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135 views

Analytical solution for an almost geometric series

Is there any way of solving explicitly the limit of the series $\sum_{n=0}^\infty q^n a^{p ^ n}$ where $0<p,q<1$ and $a>0$? The series is obviously convergent as $a^{p ^ n} < ...
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0answers
1k views

Notation for a subsequence of a sequence

If we have a sequence (an ordered list) $$ S=(s_0,s_1,...,s_n). $$ What is the notation for expressing that $S'$ is a (ordered) subsequence of $S$?
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181 views

Can we confirm $\sum_x{x!}$?

According to Wikipedia's article on indefinite sums, they list the following formula near the bottom of the page: $$\displaystyle \sum_x{\Gamma(x)}=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}{e}+C$$ ...
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107 views

Combinatoric Sequences

I have a combinatorial problem that I think is related to sequences over an alphabet. The situation is the following: I have an alphabet of n symbols and I want to look at sequences that contain each ...
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0answers
497 views

Finding a radius of convergence

Let $\sum_0^{\infty} a_n z^n$ have radius of convergence $R$ with $0< R< \infty$. Let $\alpha>0$. Find the radius of convergence of $\sum_0^{\infty} |a_n|^{\alpha} z^n$. I tried to start ...
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0answers
128 views

Perturbations of a function of bounded variation

Suppose $f,g:[0,1]\rightarrow[0,1]$, $g>0$, and $f/g$ is of bounded variation. If $f_n, f \in C^1[0,1]$ and $f_n \rightarrow f$ in $C^1$. Does it follow that $\exists N$ such that $f_n/g$ is of ...
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141 views

Has anyone studied this particular sequence?

I'm going to tag this as reference request, since I'm mainly interested in finding out whether this kind of sequence has been named in literature before, just in order to acknowledge it for something ...
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0answers
72 views

Relating product integrals to indefinite products

The product integral is the multiplicative version of standard integrals. Indefinite products are the discrete counterpart to this integral; they multiply iterations on a function $f(x)$ by each ...
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0answers
27 views

Can you estimate the difference of primes between numerator and denominator?

Let $p_n$ the nth twin prime, it is $p_n$ is a prime number and $2+p_n$ is also a prime. It is well know that Brun's theorem states (unconditionally) that $$\mathcal{B}=\sum_{n\geq ...
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0answers
30 views

Derive an inequality using Summation by Parts

Can someone help me to derive the following inequality using Summation by Parts? $a_n$ is a decreasing sequence of positive terms. $$\left|\sum_{k=m+1}^{n+p} a_k \sin kx\right| \le ...
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27 views

Find minimum value of $n$ for an integer $A$ such that $A=n^x$,where $n>1$ and $x\geq 1$

How can I calculate sum of a series of function $f(A)$ for $A = 2,3,4,5,6...A$ $f(A)=n$ (such that $n$ is minimum integer such that $A=n^x$ where $n>1$ and $x≥1$ and both n and x are integer) ...
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0answers
22 views

Difference Equations and Discrete Integral and Derivative

So I'm trying to learn difference equations, and the book that I'm using defines the following: The discrete derivative of a function $a_n$ of the integers is defined as: $$ D a_n = a_{n+1} - a_n $$ ...
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32 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
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0answers
21 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
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22 views

Denominators of harmonic numbers: asymptotic behaviour.

About the sequence $d_n$ of the denominators of harmonic numbers, I know these facts: It is unbounded, since $p\mid d_p$ for any prime $p$. It contains only one $1$. What more is known? Specially, ...
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0answers
20 views

Infinite series, continued fractions, nested radicals and such - is there a general recursive algorithm theory?

The nature of infinity and irrational numbers (among other things) always gave me trouble. But recently I learned to think about infinite sequences in terms of recursive algorithms and their time ...
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0answers
14 views

Ratios between index and sequence element

Let $a_1\le a_2\leq\ldots$ be an infinite sequence of positive integers. A positive integer $n$ is called good if $i=na_i$ for some index $i$. For which $(m,n)$ is it true that if $m$ is good, then ...
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0answers
20 views

Implications of convergence in distribution

I would like to ask you for an help to show $(\star)$ Consider a sequence of real-valued random variables $\{X_n\}_n$ and assume $\sqrt{n}(X_n-\mu)\rightarrow_d (N,\sigma^2)$ as $n \rightarrow ...
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0answers
19 views

When the supremum of a real sequence is finite and not attained, it coincides with the limsup

I'm having a bit of a problem with an exercise I have to make. In the exercise we are given the sequence $(s_n)$, which is a sequence of reals. Furthermore, we are given that $m=\sup\{s_n|n \geq ...
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0answers
32 views

Different representation of $f(n) = \sum_{d|n; \ \sqrt n\le d \le n}(-1)^d$

I am looking for a different way to calculate the following sum where $d,n\in \mathbb N$: $$f(n) = \sum_{d|n; \ \sqrt n\le d \le n}(-1)^d$$ Here are some example results for different values of n ...
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0answers
25 views

Show $\limsup S_k=y$ and $\liminf S_k=x$ from rearrangement of series

I'm in a process of proving the last part of Riemann's Theorem on conditionally convergent series. The theorem states: Let $\sum a_n$ be a conditionally convergent series with real-valued terms. ...
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0answers
52 views

Why the differential equations have a wave behavior?

The differential equation for string: $$\frac{1}{c^2} \frac{\partial^2 f}{\partial t^2}=\frac{\partial^2 f}{\partial x^2} \tag{1}$$ I have inital condition: $$f(x)=\begin{cases}20x, & 0\le ...