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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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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 ...