8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
517
(i) Substitute x 2 for x in the Taylor series (8.24). Write out the first several
terms and compare to your work in part (a). Explain why the substitution
in this problem should give the Taylor series for sin(x 2 ) centered at 0.
(ii) What should we expect the interval of convergence of the series for sin(x 2 )
to be? Explain in detail.
4. Based on the examples we have seen, we might expect that the Taylor series for a
function f always converges to the values f (x) on its interval of convergence. We
explore that idea in more detail in this exercise. Let f (x) =
e −1/x 2 if x 0,
0
if x = 0.
(a) Show, using the definition of the derivative, that f ′ (0) = 0.
(b) It can be shown that f (n) (0) = 0 for all n ≥ 2. Assuming that this is true, find
the Taylor series for f centered at 0.
(c) What is the interval of convergence of the Taylor series centered at 0 for f ?
Explain. For which values of x the interval of convergence of the Taylor series
does the Taylor series converge to f (x)?
517
(i) Substitute x 2 for x in the Taylor series (8.24). Write out the first several
terms and compare to your work in part (a). Explain why the substitution
in this problem should give the Taylor series for sin(x 2 ) centered at 0.
(ii) What should we expect the interval of convergence of the series for sin(x 2 )
to be? Explain in detail.
4. Based on the examples we have seen, we might expect that the Taylor series for a
function f always converges to the values f (x) on its interval of convergence. We
explore that idea in more detail in this exercise. Let f (x) =
e −1/x 2 if x 0,
0
if x = 0.
(a) Show, using the definition of the derivative, that f ′ (0) = 0.
(b) It can be shown that f (n) (0) = 0 for all n ≥ 2. Assuming that this is true, find
the Taylor series for f centered at 0.
(c) What is the interval of convergence of the Taylor series centered at 0 for f ?
Explain. For which values of x the interval of convergence of the Taylor series
does the Taylor series converge to f (x)?