Can these power series be used to manually compute the sine or cosine of some weird angle like π/5 for instance that none of the sum or difference identities work on? Definitely makes Fall 2024 (and next semester by extension, since 3A is a prerequisite for 3B) all the more exciting if so.
Some strange functions do not have power series representations, but most of the ones we work with on a regular basis do. A function that has a power series is called an analytic function. One of the requirements for a function to be analytic (have a power series) is that it has to be infinitely differentiable, as you might guess from the process in the video which involves taking derivatives iteratively. The good news is that functions like the trig functions, exponential, logarithmic, hyperbolic, polynomial, and finite algebraic combinations of those things (sums, products, etc) are all analytic on their domains.
Can these power series be used to manually compute the sine or cosine of some weird angle like π/5 for instance that none of the sum or difference identities work on? Definitely makes Fall 2024 (and next semester by extension, since 3A is a prerequisite for 3B) all the more exciting if so.
So, you're saying that we could write all of the functions as a particular power series? weird!
Some strange functions do not have power series representations, but most of the ones we work with on a regular basis do. A function that has a power series is called an analytic function. One of the requirements for a function to be analytic (have a power series) is that it has to be infinitely differentiable, as you might guess from the process in the video which involves taking derivatives iteratively. The good news is that functions like the trig functions, exponential, logarithmic, hyperbolic, polynomial, and finite algebraic combinations of those things (sums, products, etc) are all analytic on their domains.