dwsmith said:What would be the power series of $g(t)$?
$$
g(t) = \sum_{n=0}^{\infty}\frac{g(t)}{n!}
$$
This?
A power series is a mathematical series that is used to represent a function as an infinite sum of terms. It is typically written in the form of a polynomial with infinitely many terms, where each term has a variable raised to a different exponent.
The radius of convergence for a power series is found by applying the ratio test. This involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If this limit is less than 1, the series converges, and the radius of convergence is equal to the distance from the center of the series to the nearest point at which the series diverges.
A power series is a specific type of series that represents a function, while a Taylor series is a type of power series that is centered at a specific point. A Taylor series includes terms that represent the derivatives of the function at that point, while a general power series does not have this restriction.
Power series are useful in many mathematical applications, such as approximating functions, solving differential equations, and evaluating integrals. They allow for complex functions to be represented in a more manageable and understandable form.
No, not all functions can be represented by a power series. The function must be analytic, meaning that it can be represented by a convergent power series in some interval. Functions that are not analytic, such as step functions or functions with discontinuities, cannot be represented by a power series.