The following infinite series can be used to approximate e^x:

(a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion (Eq. 4.13) with x_i = 0 and h = x.

(b) Use the Taylor series to estimate f (x) = e^−x at x_{i + 1} = 1 for x_i = 0.25. Employ the zero-, first-, second-, and third-order versions and compute the |ε_t| for each case.

Transcribed Image Text:

x3 3! et = 1+x+· x+1/²+ +...+ 2 x" n!