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19 lines
1.2 KiB
Markdown
19 lines
1.2 KiB
Markdown
#math/calculus
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if [[Series#Geometric series|geometric]], find $a$ and $r$, and use the formulas.
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if algebraic:
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- try getting the partial fraction of the function $a_n$
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- begin to compute the series from beginning
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- see whether any element of the partial fraction cancels any other iteration.
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Newton's method ::: $x_n=x_n-1-f(x_1)/f'(x_1)$
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Approximation the zero of a function.
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[Newton's method - Wikipedia](https://en.wikipedia.org/wiki/Newton%27s_method) :
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If the function satisfies sufficient assumptions and the initial guess is close, then
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$$
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x_{1}=x_{0}-\frac{f\left(x_{0}\right)}{f^{\prime}\left(x_{0}\right)}
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$$
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is a better approximation of the root than $x_0$. Geometrically, $(x_1, 0)$ is the intersection of the _x_-axis and the [tangent](https://en.wikipedia.org/wiki/Tangent "Tangent") of the [graph](https://en.wikipedia.org/wiki/Graph_of_a_function "Graph of a function") of $f$ at $(x_0, f(x_0))$: that is, the improved guess is the unique root of the [linear approximation](https://en.wikipedia.org/wiki/Linear_approximation "Linear approximation") at the initial point. The process is repeated as$$
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x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}
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$$ functions and to systems of equations. |