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8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
Activity 8.26.
Let P n (x) be the nth order Taylor polynomial for sin(x) centered at x = 0. Determine
how large we need to choose n so that P n (2) approximates sin(2) to 20 decimal places.
⊳
Example 8.5. Show that the Taylor series for sin(x) actually converges to sin(x) for all x.
Solution. Recall from the previous example that since f (x) = sin(x), we know
f
(n+1) (x) ≤ 1
for any n and x. This allows us to choose M = 1 in the Lagrange error bound formula.
Thus,
|P n (x) − sin(x)| ≤
|x| n+1
(n + 1)!
(8.23)
for every x.
We showed in earlier work with the Taylor series
∞
k=0
x k
k!
converges for every value of x.
Since the terms of any convergent series must approach zero, it follows that
lim
n→∞
x n+1
(n + 1)!
= 0
for every value of x. Thus, taking the limit as n → ∞ in the inequality (8.23), it follows
that
lim
n→∞
|P n (x) − sin(x)| = 0.
As a result, we can now write
sin(x) =
∞
n=0
(−1) n x 2n+1
(2n + 1)!
for every real number x.
Activity 8.27.
(a) Show that the Taylor series centered at 0 for cos(x) converges to cos(x) for
every real number x.
(b) Next we consider the Taylor series for e x .