0
votes
0answers
30 views

Superlinearly convergent

A sequence $\{p_n\}$ is said to be superlinearly convergent to $p$ if $$\lim_{n\to \infty}{\frac{|p_{n+1}-p|}{|p_n-p|}}=0$$ a. Show that if $p_n\to p$ of order $\alpha$ for $\alpha>1$, then ...
3
votes
1answer
155 views

Estimating linearly independent solutions to third order recurrence relation

I'm trying to prove something about two linearly independent solutions; $a_n$, $b_n$, to a recurrence relation I have - specifically that $\left| \frac{a_n}{b_n} \right|$ is eventually monotonically ...
0
votes
0answers
39 views

Show an iterative method converges and find the rate of convergence - Numerical Analysis

We are defining an algorithm as follows: Let $f(x)$ be a function with a root in $[a,b]$. We define a series $\{x_k\}_{k=1}^{\infty}$ as follows: $x_{k+1}=x_k-f(x_k)\frac{b-a}{f(b)-f(a)}$. Does ...
1
vote
1answer
22 views

Find $R$ in $ B = \sum_{i=1}^{60}P_i\left(1+R\right)^{60-i} $ given $B$ and $P_1, P_2, …, P_{60}$

I don't have much training in financial math and was unsure of where to start tackling the following... Given a series of payments $P_1, P_2, ..., P_{60}$ which are increasing sporadically from one ...
3
votes
0answers
86 views

Loss of Significance problems - Taylor Expansion

(2) This question addresses the notion of loss of significance. You are encouraged to revisit the Taylor series expansion that you have learned in calculus, as you will need to apply it here. Explain ...
1
vote
0answers
19 views

nonlinear sequence to sequence transformations

i know matrix methods such as Cesaro,Holder,Riesz are regular linear sequence transformations. i wonder if there is any regular nonlinear sequence transformation?
1
vote
1answer
95 views

In EMI calculations, how to calculate “Rate” if EMI, Principal and Time are given

In EMI (Equated Monthly Instalments) calculations, the inputs are- Principal-P, Rate-r, and ...
3
votes
1answer
46 views

Rate of convergence for series.

1)What is convergence rate of a series \begin{equation} K(k) = {\frac{\pi}{2}} \hspace{1mm} {\sum_{m=0}^\infty}\binom{-1/2}{m}^2 k^{2m} \end{equation} Note that the presence of squares of ...
0
votes
1answer
34 views

experimental sequence of number

I'm doing a small numerical experiment. I got, from the first simulations, the following sequence of numbers. I'm trying to imagine a mathematical law behind this sequence. It could be a geometric ...
3
votes
1answer
89 views

Why below sequence is diverge?

This problem maybe simple for you,but i dont know that why below sequence is diverge?please help me about this: why $‎\lbrace\mid x_{k}\mid‎\rbrace$ with below definition is diverge? $x_{k+1}:= ...
2
votes
1answer
120 views

approximation of sum of gaussian-like function?

Let: $g(u; x,s) = \dfrac{1}{s\sqrt{2\pi}} \exp\left(-\dfrac{1}{2} \left(\dfrac{x-u}{s}\right)^2\right)$ Where $x,s$ are parameters I'm looking for a closed-form solution or approximation of: ...
0
votes
0answers
76 views

Convergence acceleration of series by suitable dummy variables

I asked this question in Mathoverflow but did not get any answer so I am posting it here hoping to get an answer here (usually its the other way round) We can show that if $|\frac{x}{x+y}| < 1$; ...
0
votes
1answer
131 views

How should I interpolate between values in a logarithmic series?

What's the best way to interpolate between 2 values of a logrithmic series? More specifically, I have a process where we encode values as $b = \text{floor}(\log(x, k))$. We discard the original ...
4
votes
1answer
196 views

How to prove that the following iteration process converges?

I have the following iteration process: $$ p_{n+1} = \frac{{p_{n}}^3 + 3 a p_{n} }{3 {p_{n}}^2 + a } , $$ where $a > 0$. Q1: How to prove that this iteration process converges for every number ...
1
vote
1answer
72 views

Numerical Analysis converging sequence question

Show that the sequence $p=10^{-2n}$ converges to zero with order $2$. How many steps, $n$, will it take before this sequence is within $10^{-8}$ of zero? Construct a sequence that converges with ...
4
votes
4answers
45 views

Generating a Monotonically Decreasing Sequence that adds to 1 for any length

I would like to generate a monotonically decreasing sequence whose elements will add to one, and generate this sequence for any order $L$. For instance, if $L=2$, then $\vec{s} = [ \frac{3}{4}, ...
0
votes
1answer
328 views

Proof of convergence of a telescoping series

Show that the telescoping series below converges if and only if the $\lim_{j\to\infty} c_j$ is defined and finite. $$\sum_{j=1}^{\infty} c_j - c_{j+1}$$ Not really sure where to start for ...
2
votes
1answer
92 views

Stretching of a set of numbers to align to a reference

I am trying to align an ordered set of n real, strictly positive numbers $$Q = {q_{1},q_{2},..., q_{n}}$$ to a reference set of the same size and with the same properties $$R = {r_{1},r_{2},..., ...
4
votes
0answers
142 views

Inexact Newton method.

Let's a nonlinear function $ f:[-1,+1]^N\subset\mathbb{R}^N\to\mathbb{R}^N,\; N\in\mathbb{N}, $ such that the the sequence generated by the method of Newton-Raphson $$ x_{n+1}=x_n-[Df(x_n)]^{-1}\cdot ...
6
votes
2answers
290 views

Stirling's Series Derivation

I was reading this paper and at the top of page 9 it says that as $n\to\infty$, $$\left(1+\frac{1}{n}\right)^{n+1/2}e^{-1}\left(1+\frac{a_1}{(n+1)}+\frac{a_2}{(n+1)^2}+\cdots ...
0
votes
1answer
122 views

Application of Backward euler method

Information Let $y'(t)=f(t,y(t))$ and $y(0)=y_0$ The backward euler method together with the center rule is given by: $y(t_k)=hk$ where $h\in (0,\frac{1}{K})$ is the step size. Recursion: ...
0
votes
1answer
76 views

Approximating a simple sum

Can somone help me find an assymptotic formula for n, for fixed x , for this sum , perhaps an inequality would be even better, or some bound on the error. $$\sum_{k=1}^n \frac{1}{\log(kx)}$$ I need ...
1
vote
1answer
2k views

Relationship between rate of convergence and order of convergence

What is the difference between rate of convergence and order of convergence? Have they any relationship to each other? For example could i have two sequences with the same rates of convergence but ...
1
vote
1answer
71 views

How do I show that the limit is to the 600th places and is it too hasty to take the limit?

I am a little unsure about (b) and (d) For (b), is it appropriate to just take the limit directly? (do I have to show that the sequence is bounded and monotone?) from $x_{n+1} = ...
1
vote
0answers
33 views

Constructing a sequence [duplicate]

Possible Duplicate: Terms of a Sequence Construct a sequence of interpolating values $Y_n \text{ to }f(1 + \sqrt{10})$, where $f(x) = (1 + x^2)^{-1}$ for $-5 \leq x \leq 5$, as follows: ...
3
votes
3answers
122 views

Accuracy from approximating $\zeta(2)$ with a partial sum

This is for an introductory numerical analysis class. The answer shouldn't be too complicated, but if you have one, feel free to post it. Figure out what $n$ should be such that $$\sum_{k=n+1}^\infty ...
2
votes
1answer
175 views

Is the Neumann series of a real semidefinite matrix asymptotic?

Let $A \in \mathbb{R}^{m \times m}$ be a real symmetric negative-semidefinite matrix and consider the Neumann series $$\sum_{k=0}^\infty t^kA^k = I + tA + t^2 A^2 + \cdots$$ where $t > 0$ is a ...
1
vote
0answers
372 views

solving Bessel function equation by hand

I have a Bessel function of the first kind given by the equation $$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m+\alpha}$$ I am trying to ...
4
votes
4answers
612 views

Calculation of Bessel Functions

I want to calculate the Bessel function, given by $$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m}$$ I know there are some tables that ...
38
votes
10answers
2k views

What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
1
vote
2answers
95 views

Calculate breakeven when fixed revenue being added per month v/s cost

I am writing up a cost sheet for a product and I basically suck at math. Didn't know who else to turn to, so trying out Math exchange. So, I am planning to spend ...
3
votes
1answer
117 views

Infinite Series: Practical Termination Criteria

I want to take partial sums of a convergent series. But how will I know how accurate the approximation will be? I am looking for practical ways of achieving this even if not very rigorous! Suppose ...
4
votes
1answer
270 views

Series expansion for iterated function

I would like to find the MacLaurin expansion of an iterated function. Finding the first few terms is not hard, but it doesn't take long before Mathematica runs out of memory using the straightforward ...
6
votes
1answer
432 views

series including infinite sum

I am looking for the approximation of the following function: $$\rho(a,b)=1-e^{-(a+b)}\sum_{m=1}^{\infty}\left(\sqrt{\frac{b}{a}}\right)^m I_m(2\sqrt{ab})$$ where $I_m(x)$ is the modified Bessel ...
6
votes
2answers
545 views

Lagrange Inversion of power series for fractional exponents?

I understand how they obtained the inversion of sin(x) shown here, using the Lagrange Inversion Formula, and have even written a MATLAB script to solve the inversion when input and output exponents ...
1
vote
1answer
846 views

How can I find the constant of integration?

I'm attempting a novel approach to some tough integration problems. I'm using the idea of series expansions to help integrate. In other words, I will attempt to approximate integration by ...
6
votes
2answers
377 views

Accelerating Convergence of a Sequence

Suppose I had a monotonically increasing sequence $\{d_{n}\}$ which is also bounded above. The $d_{n}$'s satisfy a given recurrence, however computationally they tend very slowly to the limit. What ...
5
votes
0answers
120 views

Shintani cone zeta function

Is there a procedure/algorithm for calculating sums of the form $$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$ where $$ L_i(n_1,\ldots, n_r) ...
2
votes
0answers
401 views

Numerical Methods: Approximating an Integral with Exponentials [closed]

I'd like to know about the best numerical methods for approximating an integral. Unfortunately, I want to know about a fairly general case, so I cannot give a lot of information. Essentially, I have ...
6
votes
1answer
933 views

Levin's u-transformation

Suppose I'm given a very slowly converging sequence $\sum_k a_k$. In the literature, the Levin u-transformation is mentioned as a good universal technique for convergence acceleration. I have ...
1
vote
1answer
724 views

Second order Taylor method to solve system of equations

How do I use second order Taylor method to solve a system of non-linear equations? Is there a good reference that gives details? I found mentions of it in a dozen of numerical analysis books, but no ...
2
votes
3answers
187 views

Determine speed of the object at the current time by the non-uniform time sample

Here is a time sample: $Q = \{(t_i, x_i) | 0 \leq x_i \leq x_{i+1}, 1 \leq i \leq n\}$ and rules: (1) $T_1 \leq t_{i+1} - t_i < T_2$ where $T_1, T_2 > 0$ (2) $x_i$ comes with error: $x_i = ...