Looking at the series which function do you predict will be

Question

Question: Consider the functions

y = e^x2 and y = 1/(1 + x²)

(a) Write the Taylor expansions for the two functions about x = 0. What is similar about the two series? What is different?

(b) Looking at the series, which function do you predict will be greater over the interval (-1, 1)? Graph both and see.

(c) Are these functions even or odd? How might you see this by looking at the series expansions?

(d) By looking at the coefficients, explain why it is reasonable that the series for y = e-x2 converges for all values of x, but the series for y = 1/(1 + x2) converges only on (-1, 1).