The discussion focuses on applying Taylor series for functions where x approaches infinity, specifically for the function f(x) = x√(1+x²)(2x²/3 - 1) + ln(x + √(1+x²)). Participants emphasize that for large x, the Taylor series of f(1/x) should be evaluated around 0 to obtain good approximations. However, they note that this method may not work for divergent series, which may require the use of a Laurent series for accurate approximations. The conversation highlights the importance of understanding the behavior of functions as they approach infinity.
PREREQUISITESStudents and professionals in mathematics, particularly those studying calculus and series expansions, as well as anyone looking to understand function behavior at infinity.
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.