>>14166455I guess you need to check that
(1) The Taylor series actually converges to something for |x| < some limit
(2) This limit for the series for sin(x), for cos(x), and for exp(x) is infinity
(3) That you can differentiate one of these Taylor series term-by-term, and the result will be another convergent Taylor series that is the derivative of the original series
(4) That the TS for exp(x) is its own derivative
(5) That the derivative of a^x with respect to x, taken over x in the rationals, is c(a)*a^x for some function c(a), which we can call the natural logarithm of a.y
(6) That we can find a value e, so that c(e)=1. Then the derivative of e^x is e^x itself.
(7) That the differential equation f'(x)=f(x), subject to the constraint f(0) = 1, has a unique solution. So the Taylor series given for exp(x) does give the value for e^x.
(A) But Taylor series don't always converge for all x, and don't always evaluate to the right values even when they do converge. There is a standard example of a function all of whose derivatives at 0 are zero, but isn't a constant.
