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Example: Determine the Taylor series for f(x) = ex about x=0.
Solution
It is probably one of the easiest functions to get the Taylor series for. We just require recalling that,
f(n)(x) = ex n = 0,1,2,.....
Therefore we find,
f(n)(0) = 1 n = 0,1,2,.....
The Taylor series for this illustration is then,
Obviously, it's frequently easier to find the Taylor series about x=0 but we don't all the time do that.
Definition
A function, f(x), is termed as analytic at x=a if the Taylor series for f(x) regarding to x=a has a positive radius of convergence and converges to f(x).
Proof of: lim q →0 (cos q -1) / q = 0 We will begin by doing the following, lim q →0 (cosq -1)/q = lim q →0 ((cosq - 1)(cosq + 1))/(q (cosq + 1)) = lim q
detail on identity function
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