IB Math HL Y2
Mr. Jauk
Convergence of Taylor, Error Estimates
Name
Date
Part 1: Introduction
This section addresses two questions that were left unanswered in the previous section.
1. When does a Taylor series converge to its generating function?
2. How accurately do a function’s Taylor Polynomials approximate the function on a given
interval?
We can answer these questions with the following theorem.
Taylor’s Theorem
If f and its first n derivatives f ', f ", ....... f (n) are continuous on [ a,b ] or on [ b,a ] , and f (n)
is differentiable on (a,b) or on (b,a) , then there exists a number c between a and b such that
f "(a)
f (n) (a)
f (n+1) (c)
2
n
f (b) = f (a) + f '(a)(b - a) +
(b - a) + ... +
(b - a) +
(b - a)n+1
2!
n!
(n + 1)!
When we apply Taylor’s theorem, we usually want to hold a fixed and treat b as an independent
variable. Taylor’s formula is easier to use in circumstances like these if we change b to x.
Corollary to Taylor’s Theorem (Taylor’s Formula)
If f has derivatives of all orders in an open interval I containing a, then for each positive
integer n and for each x in I,
f (x) = f (a) + f '(a)(x - a) +
where Rn (x) =
f "(a)
f (n) (a)
(x - a)2 + ... +
(x - a)n + Rn (x),
2!
n!
f (n+1) (c)
(x - a)n+1
(n + 1)!
for some c between a and x.
So, when we state Taylor’s theorem in this way, it says that for each x in I,
f (x) =
What can you conclude about this equation?
IB Math HL Y2
Mr. Jauk
Convergence of Taylor, Error Estimates
Example 1
Show that the Taylor series generated by f (x) = ex at
value of x.
x = 0 converges to f (x) for every real
Part 2. Estimating the Remainder
It is often possible to estimate Rn (x) as we just saw.
Remainder Estimation Theorem
If there are positive constants M and r such that f (n+1) (t) £ Mr n+1 for all t between a and x,
inclusive, then the remainder term
Rn (x) £ M
r
x-a
(n + 1)!
n+1
Rn (x) in Taylor’s theorem satisfies the inequality
n +1
.
If these conditions hold for every n and all the other conditions of Taylor’s theorem are satisfied
by f , then the series converges to f (x).
IB Math HL Y2
Mr. Jauk
Convergence of Taylor, Error Estimates
Example 2.
(a) Show that the Maclaurin series for cos x converges to cos x for every value of x.
(b)Hence find the MacLaurin series explansions for the following, including their radii of
convergence. i) cos(2x) and then ii) sinx
IB Math HL Y2
Mr. Jauk
Convergence of Taylor, Error Estimates
Example 3: Let 𝑓(𝑥) = 𝑙𝑛(1 + 𝑥)
(a) i) Find the MacLaurin series for 𝑓(𝑥) and its associated interval of convergence I.
ii) Show that 𝑓(𝑥) = 𝑙𝑛(1 + 𝑥) equals its MacLaurin series for 0 < 𝑥 ≤ 1 by showing
that lim 𝑅𝑛 (𝑥: 0) = 0 𝑓𝑜𝑟 0 < 𝑥 ≤ 1
𝑛→∞
1
b) Use the geometric series to write down the power series representation for 1+𝑥 , |𝑥| <
1
IB Math HL Y2
Mr. Jauk
Convergence of Taylor, Error Estimates
Example 4: Use the MacLaurin series expansion for 𝑒 𝑥 to write down power series expansions
for
𝑖) 𝑒 2𝑥
ii) 𝑒 4𝑥
iii) 𝑒 𝑛𝑥
b) Use he macLaurin series expansion for cosx to find the first two non-zero terms of the
MacLaurin series expansion of cos(𝑒 𝑥 )
c) Hence, show that for x near 0, cos(𝑒 𝑥 ) ≈ cos(1) − sin(1) 𝑥
d) Find, by direct calculation, the Maclaurin polynomial of degree 2 of cos(𝑒 𝑥 )
IB Math HL Y2
Mr. Jauk
Convergence of Taylor, Error Estimates
Part 3. Truncation Error
x
x
The Maclaurin series for e converges to e for all x. But we still need to decide how many
x
terms to use to approximate e to a given degree of accuracy.
Example 5.
-6
Calculate e with an error of less than 10 .
Example 6.
æ x3 ö
with an error magnitude no greater than
è 3! ÷ø
For what values of x can we replace sin x by x - ç
3 ´ 10-4 ?