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

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44 views

Prove that the following sequence goes to 0

We are given a sequence of points in $\mathbb{R}_+^n$, $(\vec x_T)_{T = 1}^\infty$ that converges to a point $\vec y$. We now observe the sequence $$D_T = \frac{\sum_{t =1}^T C^{k^t}f(\vec ...
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29 views

What will be the value of a

In the sum $\sum\limits_{f(a+ib)=0}\sum\limits_{n=1}^\infty \frac{g(n)sin(blog(n))}{b}(n^{-a}-n^{a-1})$ it is given that $\lim_{n \to \infty}g(n)log(n)$ is nonzero . I applied term test to the first ...
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9 views

optimum lengths for a gauge block set

Has there been any mathematical study of the "optimum" lengths for a gauge block set? What do you call such a set of lengths? I'm looking for something analogous the way different ways of ...
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27 views

About a specific mathematical series which is a power of the exponential function

My professor wrote the below exponential function just out of the box when he suggested a kernal for a 1D domain. $f(x) = e^{-\Big(a_1x_1+ \dfrac{1}{2} a_2 x_2^2 + \dfrac{1}{3} a_3 x_3^3 + .... ...
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19 views

Let $\sum_{n=0}^{\infty} a_n = M$ be a convergent series with $a_n \ge 0 \in \mathbb R$. Non-contradiction proof: $s_k \le M$ for $k \in \mathbb N$.

Let $\sum_{n=0}^{\infty} a_n = M$ be a convergent series with $a_n \ge 0 \in \mathbb R$. I want to prove $s_k \le M$ for $k \in \mathbb N$. I could do a proof by contradiction as follows: Suppose ...
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64 views

What was Euler's misconception about functions and infinite series?

I just read this on Strichartz' The way of Analysis: [...] Euler - the leading mathematician of the eighteenth - developed all techniques needed for the study of Fourier series, but he never ...
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14 views

Order of a sum, polynomial and exponential terms

I would like to understand if the following sum, $$\sum_{x=0}^L x^{d-1} p ^ {L-x}$$ is of order $L^{d-1}$ or of order $L^{d}$. In the previous expression $0<p<1$ and $d$ is an integer. In ...
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35 views

Find the closed form of $S(m, n)=\sum_{j=1}^{m}\sum_{i=1}^{n}i^j$

I know that the closed form of the sum $\sum\limits_{i=1}^{n}i^j$ can be written using Bernoulli numbers $B_k$. It is the famous Faulhaber's formula: $$\sum\limits_{i=1}^{n}i^j =\frac{1}{j+1} ...
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45 views

Find all $a_i$ such that $(x_{a_1} - x_{a_2} + x_{a_3}) +\ldots + x_{a_{3k}}$ min

Given $n$ numbers $x_1, x_2, \ldots,x_n \in \mathbb{Z}$ and an integer $k \le\frac n 3$. Find $a_i$ $(i = \overline{1,2,3,\dots,3k}),\ 0 < a_i < a_{i+1} \le n$ such that: $$M = (x_{a_1} - ...
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27 views

How to define a conditional length for a sequence?

Let $\langle a_i \rangle_{i=1}^{n}$ be a sequence from $i$ to $n$. And let $x$ be a value that can vary, example: $x=100$. Which would be the best notation if I wanted to put a condition to limit the ...
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59 views

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then $A.$ a,b,c,d are in G.P. $B.$ a,b,c,d are in H.P. $C.$ a,b,c,d are in A.P. $D.$ None of the above Tried writing the expression as a ...
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36 views

Is something wrong in this proof?

Show that if $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n| > 0 $, then $R\leq 1$. Proof: Suppose that $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n|=\alpha > 0$. ...
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62 views

Using the root test twice.

Lets for example consider the following seires $\sum (something)^{n^2}$ Can I use the root test twice here and get $(something)$ then calculate the limite and then decide if the series is convergente ...
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70 views

Is there a formula for this sequence?

The following Mathematica program: ...
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22 views

Prove series of functions is uniformly convergent then $f_n \rightrightarrows 0$

I am doing some exercise on Abbott's textbook and there is one which says: Let $f_n:S \subset \mathbb R \to \mathbb R$. Prove that if $\sum_{k=1}^{\infty} f_n$ is uniformly convergent on $S$, then ...
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17 views

Expanding a Maclaurin series

I need to expand the following in a Maclaurin series: $$ \frac{r_2}{r_1} = [1 - 2(\frac{a}{r_1})sin(\theta) + (\frac{a}{r_1})^2]^\frac{1}{2} $$ to get: $$ r_2 = r_1 - asin(\theta) + ...
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14 views

If $\sum_kg_k$ converges, then does $\sum_kf_kg_k$ converge as well when $f_k=0$ for even $k$ and $f_k=\pm1$ for odd $k$?

Take a sum $\sum_kf_kg_k$. Take $f_k=0$ for even $k$ and alternates $f_k=\pm1$ for odd $k$, so that we have $\{(k,f_k)\}=\{(1,1),(2,0),(3,-1),(4,0),(5,1),(6,0),(7,-1),\ldots\}$. If we can show that ...
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23 views

Two dimention recursive recursive equation

I am unable to solve the following recursive equation which I must solve in my research problem. Please give me advice or solution to the problem. For $K=\min(N/2,C)$ and N,C T_c, T_s,p,T are ...
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45 views

Is this sequence of functions uniformly convergent?

I have the following sequence of functions: $f_n(x)=\sin^2(\pi/x)*\chi_{[1/n+1,1/n]}$. So, $f_n(x)=\sin^2(\pi/x) , x \in(1/(n+1),1/n)$ or $f_n(x)= 0, x \notin(1/(n+1),1/n)$. I thing the sequence is ...
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25 views

I've managed to find a way to swap my sms around, but I don't know how to take the limit (and what it is valid for)

This is self learning, without a textbook, this is entirely my own and without the stabalisers of a textbook I'm worried Earlier today I posted I cannot see what happens to ...
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31 views

Recurrence in Two Variables

Anyone know how to solve the following recurrence relation in two variables: $$ f(x,y) = b f(x-1,y) + c f(y,x-1) \\ f(x,0) = b^{(x-1)} \\ f(0,y) = 0 $$ (Note: repost of a post I asked yesterday with ...
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21 views

If C is a non-empty closed set in $\mathbb{R}^{N}$ and $x \in \mathbb{R}^{N}$, prove there is a point C closest to z

The hint I've been given is to choose a sequence of points in C whose distances from x converge to the infimum. And if C is closed and bounded, how can I prove there is a point of C furthest from x? ...
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30 views

Practical methods to prove uniform convergence?

Can you please suggest some practical methods to prove whether a series of function (resp a sequence) is point-wise or uniformly convergent?
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122 views

Proving existence of uniformly convergent subsequence of a sequence of functions

Let $\{f_n\}_{n \in \mathbb N}$ a sequence of integrable and uniformly bounded functions $f_n:[a,b] \to \mathbb R$ and for each $n$ let $F_n:[a,b] \to \mathbb R$ such that $F_n(x)=\int_a^x f_n(t)dt$ ...
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25 views

Help Establishing Divergence of Series (Apostol, *Calculus* Vol. I, Section 10.14 #11)

I'm having trouble establishing the divergence of the series $\sum_{n = 2}^{\infty} a_n$ where $a_n = \log^{-s} n$. The trouble is that while I can establish divergence quite easily for $s \leq 1$, I ...
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52 views

$x_{n+1} = x_n - \frac{1}{k}x_n + \frac{a}{kx_n^{k-1}}$

Be $ a, x_0 \in \mathbb{R}, ~~~ a >0, ~~~ x_0 >0 $ and be $ k \in \mathbb{N},~ k \geq2 $ The sequence is defined as follow: $$x_{n+1} = x_n - \frac{1}{k}x_n + \frac{a}{kx_n^{k-1}} \qquad n \in ...
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42 views

Radius of convergence of $\sum_{n\ge 1}{\frac{x^{n^2}}{n^2}}$

I want to determine the radius of convergence of the power series $$\sum_{n\ge 1}{\frac{x^{n^2}}{n^2}}$$ Is my following try correct, and is there any simpler way to do this: Put ...
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60 views

Finding interval convergence $\sum\frac{x^k}{k^k}$

How would I find the interval of convergence for $\sum\frac{x^k}{k^k}$ I did the ratio test. $\frac{x^{k+1}}{(k+1)^{(k+1)}}$*$\frac{k^k}{x^k}$ $\frac{x k^k}{(k+1)(k+1)^k}$ I got ...
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18 views

Simplified Form for a Series

Is there a simplified form for the following sum: $Z = \frac{r_1}{c}+\frac{r_2}{c+r_1}+\frac{r_3}{c+r_1+r_2}+...+\frac{r_n}{c+r_1+...+r_{n-1}}$ I need to express it if possible in a way that I can ...
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24 views

Hermite function

What is the Hermite function representation of the following Confluent Hypergeometric functions? $$a_0 \ _1F_1({1+\lambda\over 2},{1\over2},{-Z^2 \over D})$$ $and$ $$a_1z\ _1F_1({2+\lambda\over ...
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26 views

Evaluating the sum

Can anybody evaluate the following sum for me $ \sum\limits_{n=2}^\infty(-1)^n(\frac{(\psi(n)}{n}-\frac{\Lambda(n)}{2n}$ where $\psi(n)$ is the chebyshev function and $\Lambda(n)$ is Von mangoldt ...
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35 views

Discuss the following for the Sequence of functions :

I have come across the following series $f(x)=\sum_{n=1}^{\infty} \frac{1}{1+n^2x}$ for all $x\in R$ I want to determine the values of $x$ when the series converges absolutely and I am stuck when ...
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46 views

Asymptotic (in)dependence of a maximum of an i.i.d. sequence of Gaussian random variables on a single random variable in this sequence

Suppose that I have a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$ where $X_i\sim\mathcal{N}(0,1)$. Denote the maximum of this sequence by $M_n=\max(X_1,\ldots,X_n)$. I ...
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45 views

An identity with Big O and little O notation

For $0<\alpha \leq 1/2$, I want to show that $$\sum_{n=2}^{N}O\left(\frac{1}{n^{3\alpha}}\right)=\frac{-o(1)}{2}\sum_{n=2}^{N}\frac{1}{n^{2\alpha}} $$ Since ...
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63 views

Closed form for the following sum

I found this sum in an old math problems book and it asks me to find its closed form. And for the life of me I can't find. Here it is ...
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38 views

Hermite Functions and Power series

How to represent this series in terms of Hermite polynomials? $$y(x)=b_0\sum_{k=1}^{\infty}\left({-2 \over D } \right)^k \left[\frac{\Pi_{i=1}^{k}(2i-1+ \lambda) ...
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18 views

Product of Consecutive Terms of a Geometric Sequence

Suppose $a_n=aq^n$, where $a>0$ and $q>0$. So $(a_n)_{n=1}^\infty$ is a geometric sequence with positive terms. The product of its consecutive terms, say, $$a_0,a_2,\ldots,a_n$$ equals to ...
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78 views

Infinite sum over primes

I want to compute the following sum over primes: $$\sum\limits_{p \text{ prime}}\sum\limits_{k=1}^\infty(\log(p^k))\left(\frac{1}{2p^k} - \Phi[-1,1,p^k]\right),$$ where $\Phi[z,s,a]$ is the ...
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33 views

Limits of expectation of a function

Suppose that $g(x,y)$ is a smooth function with respect to $x$ and $y$, and that it is bounded on the domain of interest: $a\leq g(x,y)\leq b$, with $a$ and $b$ being real constants. Now let $(X)_n$ ...
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18 views

Proof that Newton expansion over derivatives has the properties of an integral

Let's consider a Newton expansion over consecutive derivatives of a function: $$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ Can it be proven that such ...
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27 views

Natural functions and their consequences

Let's define a natural function as a continuous function that is equal to its Newton expansion: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)=\sum_{m=0}^{\infty} \binom {x}m ...
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63 views

Help for applying Hahn Banach theorem in this exercise .

I want to use this version of Hahn Banach: $X$ real vector space, $M$ a subspace, $p$ a sub-linear mapping and $T:M\rightarrow\mathbb{R}$ linear and $T(x)\leq p(x)$ $\forall x \in M$ Then ...
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30 views

Limits in a topological space

Consider the topological space $(\Bbb R^2, \tau)$ described by the following neighbourhood base: $\beta_{(x,y)}= \{(x-\epsilon,x+\epsilon) \times (y-\epsilon, y+\epsilon)\subseteq \Bbb R^2: ...
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123 views

Gauss' Summation Trick; Applications and Generalizations

I'm going to write an article about the summation trick attributed to Guass and its applications and generalizations. I'm sure you know what is the trick I mean: $1+2+\cdots+100=101+101+\cdots+101$ ...
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23 views

String satisfying the condition

Given $N$, $A_0$, $B_0$, $L_0$, $A_1$, $B_1$ and $L_1$, find a sequence S consisting only of characters '$0$' and '$1$'(a total of N characters) such that: The number of '$0$'s in any consecutive ...
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51 views

Correct notation in this case

I want to define $y_n:=(1,\frac{1}{2},...,\frac{1}{n},0,..)$(hence, a sequence) for all $n \in \mathbb{N}$. And I was wondering whether $y_n:= \sum_{i=1}^{n} \frac{1}{i} (\delta_{ip})_{p \in ...
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57 views

Sum Series Simplication

I'm trying to simplify the following formula: $$\sum_{1\leq z < z^\prime < y\leq n} \frac{k^3}{z z^\prime y} (1 + \frac{k}{x + 1}) (1 + \frac{k}{x + 2}) \ldots (1 + \frac{k}{z - 1})(1 + ...
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107 views

Computation mathematics, sequences and roots

a) For $n=1,2,3..,$ let $I_n = \int_0^1 \frac{x^{n-1}}{2-x} dx$ Writing $x^n = x^{n-1}(2-(2-x))$, show that this sequence of numbers satisfies the recurrence relation: $I_{n+1} = 2I_n - ...
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65 views

What makes a pattern in a sequence?

Assume a stream of characters$(c)$ where $ c \in (A, B, C ... Z)$. I need to identify patterns available in the stream. As per the definition of a pattern I should be looking for a recurring string. ...
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110 views

How do I tell what convergence test to use for sequences and series?

So, I've been doing sequences and series for the past while in class now, and have gotten onto the test of convergence in them. However I'm currently having trouble with two major points. 1. What ...