# 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.

17 questions
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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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### 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,...
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 ... 2answers 3k views ### 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 $My question for the newtons method is what ... 1answer 61 views ### Newton method local convergence under Hölder-continuity I have to prove the following remark, but I have no idea how. I searched everywhere but didn't find anything. Remark 1: If$\nabla{F}$is only Hölder-continous with exponent$\gamma$(instead of ... 1answer 102 views ### What does$\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]...
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 ...