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Questions tagged [newton-series]

Questions regarding the Newton series expansion of functions or other finite-difference based series expansions. Issues including convergence, calculation of coefficients, and bounds on remainders.

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Estimating the Newton binomial

Could anyone help me with a exercise? I have to prove that ${n\choose s} \leqslant \left( \frac{ne}{s} \right) ^{s}$ for all $n, s \in \mathbb{N}$. Thanks for all the help
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Newtons Binomial Theorem identity

I have seen a lot of identites being discussed here but I still haven't seen the one I'm having a problem with. I need to conclude that $\sum\limits_{i=1}^{n} i\binom{n}{i} \frac{(-4)^{i-1}}{5^{i-n}} ...
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Rearranging matrix/vector multiplication (system of non-linear equations)

I am trying to solve systems of non-linear equations by hand using newtons method in preparation for an exam. Issue is that we have only used MatLab so far and no idea how to do this by hand. ...
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Convergence of damped Newton’s method [closed]

Let $f$ be a twice continuously differentiable function satisfying $LI \succeq \nabla^2 f \succeq mI$ for some $L > m > 0$ and let $x^*$ be the unique minimizer of $f$ over $\Re^n$. Proof that ...
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polynomial approximation around the point

I'm trying to figure out how to approximate a function by polynomial around the point. I'm aware of Newton's series and it is almost what I need, but it uses either forward or backward differences, ...
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Necessary and sufficient conditions for the existence of the Newton Series of a function $f: \mathbb{N} \longrightarrow R$

I’m wondering if a function $f: \mathbb{N} \longrightarrow R$ can be represented as a Newton series given that all its forward differences exist. The first thing I searched up was a result in complex ...
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Calculate the sum $\sum_{k=0}^d\binom{n+k}{m}$

Any way to calculate this sum combinatorially/analytically, pls? I known the answer is $\binom{n+1+d}{m+1}$. However I couldn't prove it.
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Polar Coordinate clarification: Newton's equations

A force can be expressed as $F=f(r)e_r$ is called a a central force. Show that the angular momentum $J(mr^2d \theta /dt$ in this case) is conserved. NOTE: $a(t) = [\frac{d^2r}{dt^2} - r(\frac{d\...
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Rate of Convergence of Function F(x) = f(x)/f′(x) using Newtons Method

Derive a formula for Newton’s method for the function $F(x) = f(x)/f′(x)$, where $f(x)$ is a function with simple zeros that is three times continuously differentiable. Show that the convergence of ...
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Newton's Method Reversed (using iteration formula to figure out f(x))

The iteration formula $x_{n+1} = x_n − \cos(x_n)\sin(x_n) + R\cos^2x_n$ , where $R$ is a positive constant, was obtained by applying Newton's method to some function $f(x)$. What was $f(x)$? What can ...
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Does this Newton series describe an interesting function, if any?

I was reading about how the harmonic numbers are analogues to the logarithm in that $\displaystyle \log(x) = \int \frac{1}{x}dx$ and $\displaystyle H_x = \sum \frac{1}{1+x} \delta x$ Where indefinite ...
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Use Newton's method to approximate a root of the equation $e^{−x}=4+x$ correct to eight decimal places.

I am having trouble with this question. They usually give us something to start with but they didn't on this one. I tried just starting from one but got the wrong answer and then tried with two, three,...
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Show that an iterative method converges to sqrt(a)

Okay, i think i solved it but it seems too easy: $\lim_\limits{x\to \infty}$ $x_n = \lim_\limits{x\to \infty}$ $x_{n+1} = \sqrt{a}$ the iterative method is: $x_{n+1}$ = $\frac{x_n(x_n^2 + 3a)}{3x_n^2 ...
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What is stopping criteria for Newtons Method?

Use newtons method to find solutions accurate to within $10^{-4}$ for the following: $$\\x^3-2x^2-5=0,\qquad[1,4]$$ Using : $p_{0}=2.0$ $\Rightarrow $ My question for the newtons method is what ...
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Newton method local convergence under Hölder-continuity

I have to prove the following remark, but I have no idea how. I searched everywhere but didn't find anything. Remark 1: If $\nabla{F}$ is only Hölder-continous with exponent $\gamma$ (instead of ...
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What does $\Delta ^{k}$ mean?

What does $\Delta ^{k}$ mean? For example in this Newton Series: $\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}[f](a)}{k!}}~(x-a)_{k}=\sum _{k=0}^{\infty }{x-a \choose k}~\Delta ^{k}[f]...
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Applying finite difference expansion to smooth but non-analytic function

There is (//en.wikipedia.org/wiki/Taylor_series) a generalization of the Taylor series that converges to the value of the function $f(x)$ for any bounded continuous function on $(0,\infty)$ using the ...