piercebeatz said:I think I've got it: the above expression is equal to
[tex] \frac{1}{x^4 - 1} - \frac{ x^2} {x^4 - 1} = - \frac{1}{1 + x^2} [/tex]
does that look ok?
A function defined as a series is a mathematical concept where a function can be represented as an infinite sum of simpler functions, known as a series. This means that the function's output can be calculated by adding up the values of the simpler functions at different input values.
In a regular function, the output can be calculated directly from the input using a specific formula or set of rules. In a function defined as a series, the output is calculated by adding up the values of simpler functions, making it more complex and requiring an infinite number of terms to accurately represent the function.
Functions defined as series are important in mathematics because they allow for the representation of complex functions that may not have a simple formula. They also have applications in calculus, where they are used to approximate functions and solve problems involving infinite sums.
No, not all functions can be represented as a series. This is because some functions may not have an infinite sum of simpler functions that accurately represents them. Functions that do not have a series representation are known as non-analytic functions.
Functions defined as series have various real-world applications, such as in physics, engineering, and economics. They are used to model complex systems and make predictions based on the input data. In finance, series functions are used to calculate interest rates and investment returns. In physics, they are used to model the behavior of waves and electromagnetic fields.