Wait.. the 4th diferential of h(x) is
I get both this question and the answer from a book. But there are 2 things I dont understand.
Please see the image below.
Therefore the first question is solved.
And here is the interesting part.
Can't you see a pattern?
Can you guys find a shortcut so that I need not to use the y=u/v formula?
And here is something I dont have any idea of
First, there isn't a shortcut for this derivative. It's really tedious sometimes.
Second, the book missed the second power in the second derivative of h(x) but from the third on the derivative of h(x) is wrong. This should explain the error in the taylor series.
Third, every time you expand a function in a power series, or when you deal with a power series that you don't know from wich function it came from, you can look for the radius of convergence of it (it gives you valuable information). That is, in other words, where the power series are meaningful, or where you can assume that it converges to some value.
You can think about the geometric series, , and you should know that it only converges if and only if |r|<1. So 1 is the radius of convergence of the geometric series.
Anyway, D'alembert's equation is a method you use for finding the radius of convergence of a power series. From now on, take this as a definition. You can find more information about power series in this site Pauls Online Math Notes.
I think this should make clear why the 2 came from in the expression |f(x)|<2. Ok?