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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 Root Test Firstly note that we can suppose without loss of generality that the series will initiate at n = 1 as we've done for all our series test proofs. As well n
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