I've recently covered the Taylor Series in my studies and have read through several of the posts here which deal almost exclusively with specific problems and proofs but none seem to be answering a question I have:
How do I pick the point "a" around which the series is centered for fastest and most accurate results (at least in theory)?
For example if I wanted to calculate Sin(1) - radians - wouldn't it be best to pick a = 1, or maybe pi/4? Almost all the problems I have seen would just use the TS around 0. Maybe using 0 just reduces the arithmetic and gets you there just as fast but from what I've read, using the TS around 1 would be more accurate.
Now, in ensuring that my approximation is within an error epsilon, I know I have to look at the remainder term, but if I center the TS around "a" this only slightly complicates the algebra in calculating the value N to ensure my error bound.
Granted this could depend on the radius of convergence also and possibly how far the "a" is from this boundary but in the case of Sin, the ROC is infinite, right?
Thank you, Chris