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Taylor’s method
What is an example of the Taylor series method?
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One classic example of the Taylor series method is approximating the value of a mathematical function using a polynomial series expansion. Let’s say we want to approximate the value of a function f(x) around a point a using the Taylor series method. The Taylor series expansion for f(x) around a is given by:
f(x) = f(a) + f'(a)(x – a)/1! + f”(a)(x – a)^2/2! + f”'(a)(x – a)^3/3! + …
In this expansion, f'(a) represents the first derivative of f(x) evaluated at a, f”(a) represents the second derivative evaluated at a, and so on. The notation n! denotes the factorial of n.
To approximate the value of f(x), we can truncate the series after a certain number of terms. The more terms we include, the more accurate the approximation becomes. For example, if we include only the first two terms, the approximation becomes:
f(x) ≈ f(a) + f'(a)(x – a)/1!
If we include the first three terms, the approximation becomes:
f(x) ≈ f(a) + f'(a)(x – a)/1! + f”(a)(x – a)^2/2!
By using more terms, we can achieve a higher degree of accuracy in the approximation.
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