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

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69 views

Can't match boundary conditions on a perturbation series solution to a non-linear ODE?

I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation: \begin{equation} \frac{d^2y}{dt^2} + y = \epsilon ...
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84 views

Uniform convergence of function series using Dirichlet criterion?

In some exercises for an analysis course, we are asked to prove that the function series $\sum f_n$ converges uniformly on $[-1,1]$, where $$ f_n:[-1,1]\to \mathbb{R}:x \mapsto \frac{x^n sin(nx)}{n} ...
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648 views

Taylor's Formula vs. Taylor's Inequality

In my calculus book, Essential Calculus, and in class we were using Taylor's formula to approximate the remainder in Taylor polynomials but I am having a bit of trouble understanding the intuition ...
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57 views

Uniform implies pointwise convergence

I had a question to show a sequence of functions $(x_n)$ in $C[0,1]$ (equipped with a metric $d$) does not contain a uniformly convergent subsequence. $$ x_n(t) = \ n(1-nt) \ \ \ \ \ \ \forall \ ...
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52 views

Taylor's Formula and 'z' values

I have attached Taylor's Formula, an exercise problem from a section on Taylor polynomials, and the solution to this exercise. I understand part a, expanding $f$ using Taylor polynomials is the ...
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33 views

Sequence with a contraction mapping among others

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following property. There exists $c \in (0,1)$ such that for all $x,y \in \mathbb{R}^n$ it holds that $\left\| f(x) - ...
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90 views

Expectation involving a maximum of a sequence of i.i.d. Gaussians

Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation: ...
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92 views

Question about the double limit of a sequence

Let $a_{mn}$ be a double sequence. Then under what conditions do we have $$\lim_m\lim_na_{mn}=\lim_n\lim_ma_{mn}$$ In particular, if the sequence is positive, and for each fixed $n$, the limit exists ...
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48 views

Finite Sum of Zeta Functions

Let $T(n) =$ the zeta function evaluated at $x = n$. Is it possible to find the sum Sigma ($k =1$ to $n$) of $T(k)$ ? via impropal integrals? closed form? I need this to evaluate a sum involving zeta ...
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143 views

Maclaurin series of $\sin(2\pi x)$

Find the Maclaurin Series for $$f(x) = \sin ( 2 \pi x )$$ using the definition of a Maclaurin series. If $f(x) = \sum_{n=1}^\infty c_{2n+1}x^{2n+1}$, give $c_{2n+1}$: ...
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46 views

Partial limits of recursive sequence

Let $\alpha \in \Re$. Find all the parial limits of the sequence $a_n$ given: $a_1=\alpha$ $a_{2n}= \frac{a_{2n-1}}{2}$ $a_{2n+1}= \frac12 +a_{2n}$ I've tried to show that $a_n$ has two ...
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42 views

Finding every set of the five consective terms of an arithmetic sequence such that the four terms of them are squares

Question : Can we find every set of the five consective terms of an arithmetic sequence such that the four terms of them are squares? Motivation : It has been known that four consecutive terms in ...
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51 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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38 views

Fixing learning rate for gradient descent single variable

I need guaranteed convergence to local minimum given initial value in $(0,6)$. The function is $f(x) = 30-61 x+41 x^2-11 x^3+x^4$. I have $x(i+1) = x_i - \eta (4x_i^3-33x_i^2+82x_i - 61) $. What ...
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72 views

Probability of a sequence

For all $k = 1, 2, ...$, consider $f_k: \mathbb{R}^k \rightarrow \{1, 2, ..., C\}$, for some given positive integer $C$, with the following properties. For all $k$, if $(y_1, ..., y_k)$ is a ...
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33 views

introductory real analysis - convergence and series Q

Show that $\forall \epsilon > 0 $$ \exists H \in \mathbb{R} $ s.t. P>H & N>H implies $\left|1 - \displaystyle\sum_{p=2}^P\sum_{n=2}^N\left(\dfrac{1}{n}\right)^p \right| < \epsilon $ my ...
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81 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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44 views

How can I find the formula for this sequence (series?)?

Apologies for the poor explanation, I haven't done maths for so long and I'm not at all sure what you would even call a problem like this, so it's probably in the wrong forum. I'm sure it's actually ...
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82 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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128 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
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45 views

Text Generating Functions: Do they exist?

This is a little far out question, but just curious: is it even possible to have a non-high-degree-polynomial function (as in polynomial regression function) that could generate a sentence of say, 10 ...
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113 views

The remainder of Taylor (Maclaurin) series of $\cos(x)$

Something is bothering me with the remainder of the Taylor (Maclaurin) series of $\cos(x)$. The formula of $a_n$ is $(-1)^k \frac{x^{2n}}{(2n)!}$. By Leibniz Theorem, $r_n<a_{n+1}$ which is, ...
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78 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
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51 views

Proving the convergence of a product

I have become interested in taking the $n^{th}$ term of a series and evaluating a product whose $n^{th}$ term is $(1+a_n)$. After looking around I came across the following inequality: ...
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25 views

Result about Convergent Subsequences

In my lecture notes I have a proof of the following: Let $(a_{2n})$ and $(a_{2n+1})$ be subsequences of $(a_n)$. If $(a_{2n})\to l$ and $(a_{2n+1})\to l$ then $(a_n)\to l$. By definition, for ...
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66 views

Computing a sum of an infinite series via partial decomposition

Consider the sequence defined by $u_0=0$ and $u_n=\frac{4n+1}{(2n+3)(4n^2-1)}$. The exercise asks to show that the series $\sum_{n\geq 0}{u_n}$ is convergent, which is clear since $u_n\sim_\infty ...
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72 views

Sequences of 'Rayleigh-like quotients' and their minima for a symmetric positive semi-definite matrix

Let $A$ be an $N\times N$ symmetric positive semi-definite matrix with eigenvalues $0 \leq \lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_N$ and corresponding eigenvectors $u_1, u_2, \dots, u_N$. ...
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25 views

If $f$ decreases and $\sum n^{a-1}{f(n)}^a<\infty$ then $\sum 2^n (2^n)^{a-1}{f(2^n)}^a$ converges.

Let $a\in(0,1]$ and $f:\mathbb{Z}_+\to\mathbb{R}_+$ be a decrasing function such that $$\sum_{n=1}^\infty n^{a-1}{f(n)}^a<\infty.$$ Prove that $$\sum_{n=0}^\infty 2^n ...
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40 views

Financial mathematics 2

I've been given the following information: MR bob borrows 50000 now and promises to repay it in monthly installments over the next eight years. Interest is compounded monthly at 18% per year. ...
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124 views

Questions about the superfactorial function.

N superfactorial or $n\$$ is defined as - $$n\$=\prod_{k=1}^n k!$$ Then is there any asymptotic formula for this? Are there any infinite series , integrals related to this function? Is there a ...
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31 views

Sum with chebyshev function

I want to compute the value of the following sum . $$\sum_{n=1}^{+\infty}(\psi(2n+1)/(2n)-\psi(2n)/(2n)).$$ $\psi$ is the second chebyshev function. It is defined in Mathematica as $$\psi(x) = ...
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351 views

Find the minimum,maximum, infimum and supremum of sets?

If $X$ is the intersection of all the intervals $(1-\frac{1}{n^2},1+\frac{5}{n^3}]$ for $n=1$ to infinity, what is the minimum, maximum, supremum and infimum of $X$? If $Y$ is the intersection of all ...
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112 views

Find $\lim_{n\rightarrow\infty} na_n$ given that $0<a_0 <1$ and $a_{n+1}=a_n-a_n^2$ for $n\geq 0$.

I understand that no matter what value $a_0$ takes on between $0$ and $1$ that $a_1\leq \frac{1}{2}$. This has lead me to believe that for all $n\geq 1$, $b_n=(\frac{1}{n}-\frac{1}{n^2})<a_n ...
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41 views

Sequences (limit properties in $ \mathbb{R}$)

Prove the following affirmation: Let $ k \in \mathbb{N}$ and $a>0$. If $a < x_n < n^k , \forall n$, then $ \lim x_n^{1/n} = 1$ I already proved that: I) If $a>0$ then $\lim a^n =1$ and $ ...
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44 views

A question about the behavior of Dirichlet series and its derivatives

Let us consider the Dirichlet series: $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ We know that its derivative is given by: $$f'(s)=-\sum_{n=1}^\infty \frac{(\ln n) a_n}{n^s} $$ My question is: What ...
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14 views

For positive $a_n, b_{n.k}$, when is $\sum_n (a_n \sum_k (-1)^k b_{n.k}) = \sum_k ((-1)^k \sum_n a_n b_{n,k})$?

A recent question gave a proposed double series for $\pi/4$, in which all inner series were alternating, which was shown to be equal to the Leibniz series for $\pi/4$ if the order of summation was ...
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44 views

Approximating arccos(a/(a+x)) for the sake of simplfying an integral

I recently tried to evaluate $$\int e^{\beta\arccos(a/(a+x))}dx$$ (everything constant except $x$) and got a complicated answer involving a hypergeometric series with complex arguments. Can anyone ...
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27 views

Substituting one series in another provided that the values of the substituted series are in the circle of convergence of the other series.

From Boas, under theorems about Power Series. This theorem is presented without an example (or proof), and I cannot exactly understand the notion of substituting one series into another one. It is ...
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40 views

on the interval $[-1,1]$, $\sum (-1)^n {x^2+n^2\over n^3}$ is absolutely convergent?

on the interval $[-1,1]$, $\sum (-1)^n {x^2+n^2\over n^3}$ is absolutely convergent right? $f_n(x)={x^2+n^2\over n^3}$ is uniformly convergent by Dinis Theorem as they are monotone and continuos on ...
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50 views

Can this limit be proven to converge to the Logarithmic Integral?

Here's the limit: $\displaystyle\lim_{k \rightarrow 1^+}\sum_{j=0}^{\lfloor\log_k x-\log_k\mu\rfloor}\frac{k^{j+\log_k\mu}}{j+\log_k\mu}$ where $\mu=1.45136380...$, Soldners Constant. Empirical ...
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133 views

Value of sum with primes

Can anyone tell me the value of sum $\sum_p\left(\log p(\frac{1}{2p}-\psi(\frac{p+2}{2})+\psi(\frac{p+1}{2})\right)-\sum_n\frac{(\log(2^n)}{2^n}$ where $p$ ranges over prime powers and $n$ ranges from ...
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74 views

Proof of absolutely convergent sums over two indices.

In the book Concrete Mathematics (2nd) written by Ronald Graham, Donald Knuth and Oren Patashnik, they prove the next theorem. Absolutely convergent sums over two or more indices can always be ...
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81 views

Literature leading up to algebraic topology

First and foremost, allow me to apologise for the quite general and perhaps vague nature of this question and for the presence of any misunderstandings in the statements I will use. I know very well ...
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101 views

Sum involving the Digamma function.

Consider the double sum $$f(x):=\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{1}{m+\frac{1}{n+x}}-\frac{1}{m+\frac{1}{n}},$$ which converges for $x>0$. One interpretation of this sum is the measure of ...
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80 views

A question on convergent sequence of positive terms

Is the following statement true: if $\{x_n\}$ is a convergent sequence of positive real numbers then either there exists a function $f: \mathbb{N} \rightarrow [1,∞)$ such that $\space$ lim ...
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132 views

Proof of Toeplitz's theorem.

Let $a_n$ be a real sequence convergent to $a \in \mathbb{R}$. Let $t_{k,n}$ (where $1\le k \le n$) be a sequence of weights such that: $$(I) \quad \forall k \lim_{n \to \infty}t_{k,n} = 0$$ $$(II) ...
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57 views

Linear algebra function that creates decreasing product vector of original vector

For vector $y=[y_1,y_2,\dots y_n]$ , let $\gamma = \sum_{i=1}^n \gamma_i$ , and $\gamma_i(n-i+1)=y_n*y_{n-1}*\dots y_i$ so that $\gamma$ looks like $[y_1*y_2*\dots y_n, y_2*\dots*y_{n}, \dots y_n]$ ...
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156 views

Problem on convergence of Series (Rudin)

This is problem $3.11$ of Rudin's real Analysis. Suppose $a_n>0$ , $s_n := a_1+a_2+...+a_n$ and that $\sum a_n$ diverges. Prove that $\sum \dfrac{a_n}{1+a_n}:=\sum b_n$ diverges. I came across ...
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49 views

Closed form of an inhomogeneous non-constant recurrence relation

I try to find a closed form for the $f_k$'s (at least for $f_1$) satisfying the following recurrence relation for any fixed $n>0$: $f_0 = 0$, $f_n = 0$, and $f_k = \frac{n}{2k}(f_{k-1} + f_{k+1} + ...
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83 views

exponential of an operator, all to a power

I saw this almost answered here: Exponential of the differential operator (it is the unaccepted answer) What I am looking to "solve" is $$ \sum_{j=0}^d\; \left( e^{\epsilon\,\partial_x} \right)^j ...