The discussion revolves around applying Taylor series for the function f(x) = x√(1+x²)(2x²/3-1) + ln(x+√(1+x²)) in the context of large x (x >> 1). Participants explore how to derive approximations for this function as x approaches infinity.
The discussion is ongoing, with participants sharing insights on the need for a Laurent series for better approximations in this context. There is recognition of the challenges posed by divergent series and the behavior of the function as x increases.
Participants note that the function approaches infinity, raising questions about the nature of approximations and the limitations of Taylor series in this scenario. The discussion includes considerations of divergent series and the need for alternative approaches.
Nice trick, I tried it but for all derivative limit to 0 is divergent...for example 1.derivative of my f(1/x) is -\frac{1+1/x^2}{\pi x^4}. Yes I would like to examine a behaviour of this function for large x and approximate it for this case.mfb said:Well, the function goes to infinity, so every approximation should do the same.
What exactly do you want to calculate or approximate where?
In general, to get good approximations for large x, you can calculate the taylor series of f(1/x) and evaluate it around 0. That won't work with divergent series, however, where you might need a Laurent series.