The Taylor series is usually written this way (using y instead of x, though).nhrock3 said:i can't understand how the got this variation of taylor series formula
[tex]f(x+h)=\sum_{k=0}^{\infty}\frac{f^{(k)}(x)}{k!}(h)^k[/tex]
http://mathworld.wolfram.com/TaylorSeries.html
when around some point there is no x-x_0
A Taylor series with plus inside is a mathematical series that represents a function as an infinite sum of terms, where each term is a polynomial expression with a plus sign in between. This type of series is used to approximate functions and make calculations easier.
A regular Taylor series represents a function as an infinite sum of terms, where each term is a polynomial expression with a minus sign in between. The plus sign in a Taylor series with plus inside indicates that the polynomial terms alternate in sign, whereas in a regular Taylor series they all have the same sign.
The purpose of a Taylor series with plus inside is to approximate a function with a polynomial expression, making it easier to perform calculations and analyze the behavior of the function. It is especially useful when the function is difficult to evaluate directly.
The general formula for a Taylor series with plus inside is given by:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... + (-1)^n * f^(n)(a)(x-a)^n/n! + ...
Where f(x) is the function being approximated, a is the point around which the series is expanded, and f^(n)(a) is the nth derivative of f(x) evaluated at a.
Taylor series with plus inside are used in various fields of science and engineering, such as physics, chemistry, and computer science. They are used to approximate the behavior of physical systems, to solve differential equations, and to design efficient algorithms for solving mathematical problems.