34 taylor's remainder theorem ii
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Taylor's Remainder Theorem II
Taylor's Remainder Theorem II
We state the general form of the Taylor's remainder formula.
Taylor's Remainder Theorem II
We state the general form of the Taylor's remainder formula.
Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b]
Taylor's Remainder Theorem II
We state the general form of the Taylor's remainder formula.
Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b]
a( )[ ]
b
f(x) is infinitely differentiable in here
Taylor's Remainder Theorem II
We state the general form of the Taylor's remainder formula.
Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and
pn(x) =
be the n'th Taylor-poly expanded around at a.
a( )[ ]
b
f(x) is infinitely differentiable in here
Σk=0
n (x – a)k
k! f(k)(a)
Taylor's Remainder Theorem II
We state the general form of the Taylor's remainder formula.
Taylor's Remainder Theorem (General Form): Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and
pn(x) =
be the n'th Taylor-poly expanded around at a.
Then there exists a "c" between a and b such that
f(b) = pn(b) + (b – a)n+1(n + 1)! f(n+1)(c)
a( )[ ]
bc
f(x) is infinitely differentiable in here
Σk=0
n (x – a)k
k! f(k)(a)
or in full detail,
f '(a)(b – a)f(2)(a)
+ 2!= f(a) + (b – a)2f(b) ..f(n)(a)
n! (b – a)n+
Taylor's Remainder Theorem II
+ Rn(b)
or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a)f(2)(a)
+ 2!= f(a) + (b – a)2f(b) ..f(n)(a)
n! (b – a)n+
Taylor's Remainder Theorem II
+ Rn(b)
or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a)f(2)(a)
+ 2!= f(a) + (b – a)2f(b) ..f(n)(a)
n! (b – a)n+
Again, we note the following
Taylor's Remainder Theorem II
+ Rn(b)
or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a)f(2)(a)
+ 2!= f(a) + (b – a)2f(b) ..f(n)(a)
n! (b – a)n+
Again, we note the following
* the theorem also works for the interval [b, a]
Taylor's Remainder Theorem II
+ Rn(b)
or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a)f(2)(a)
+ 2!= f(a) + (b – a)2f(b) ..f(n)(a)
n! (b – a)n+
Again, we note the following
* the theorem also works for the interval [b, a]
* the value c changes if the value of b or n changes
Taylor's Remainder Theorem II
+ Rn(b)
or in full detail,
where Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(a)(b – a)f(2)(a)
+ 2!= f(a) + (b – a)2f(b) ..f(n)(a)
n! (b – a)n+
Again, we note the following
* the theorem also works for the interval [b, a]
* the value c can't be easily determined, we just know there is at least one c that fits the description
* the value c changes if the value of b or n changes
Taylor's Remainder Theorem II
+ Rn(b)
Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2
Taylor's Remainder Theorem II
Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
Taylor's Remainder Theorem II
Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, …
Taylor's Remainder Theorem II
Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
= 1P(x)
Taylor's Remainder Theorem II
Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
– = 1P(x)(x – π/2)2
2!
Taylor's Remainder Theorem II
Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
– = 1P(x)(x – π/2)2
2!
1(x – π/2)4
+ 4!
Taylor's Remainder Theorem II
Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
– = 1P(x)(x – π/2)2
2!
1(x – π/2)4
+ 4!–
1(x – π/2)6
6!..1(x – π/2)8
+ 8!
Taylor's Remainder Theorem II
Example: A. Find the Taylor-series of f(x) = sin(x) at x = .π
2sin(x)
cos(x)
-sin(x)
-cos(x)
At x = π2 , we get the sequence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
– = 1P(x)(x – π/2)2
2!
1(x – π/2)4
+ 4!–
1(x – π/2)6
6!..
or P(x) = Σ(-1)n(x – π/2)2n
(2n)!n=0
n =∞
1(x – π/2)8
+ 8!
Taylor's Remainder Theorem II
Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = ) π/2
Taylor's Remainder Theorem II
Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
π/2
Taylor's Remainder Theorem II
Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
Taylor's Remainder Theorem II
Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Taylor's Remainder Theorem II
Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Since |f(n+1)(c)| < 1,
Taylor's Remainder Theorem II
Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Since |f(n+1)(c)| < 1, we've (b – )n+1
(n + 1)! f(n+1)(c) π
2 <(n + 1)!
(b – )n+1π2
Taylor's Remainder Theorem II
Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Since |f(n+1)(c)| < 1, we've (b – )n+1
(n + 1)! f(n+1)(c) π
2 <(n + 1)!
(b – )n+1π2
Again, as n we've (n + 1)!
(b – )n+1π2 0∞,
Taylor's Remainder Theorem II
Example:
B. Describe the Taylor remainder Rn(b) and show that f(b) = P(b) for all values b. (a = )
The remainder Rn(b) = (b – a)n+1
(n + 1)! f(n+1)(c)
=
π/2
(b – )n+1
(n + 1)! f(n+1)(c) π
2
where f(n+1) may be ±sin(x) or ±cos(x) and for some c that is between a and b.
Since |f(n+1)(c)| < 1, we've (b – )n+1
(n + 1)! f(n+1)(c) π
2 <(n + 1)!
(b – )n+1π2
Again, as n we've (n + 1)!
(b – )n+1π2 0∞,
Hence Rn(b) 0 and that f(b) = P(b) for all b.
Taylor's Remainder Theorem II
Differentiation and Integration of Power Series
The Taylor series of f(x) is the only power series that could be the same as f(x).
Σk=0
∞
Differentiation and Integration of Power Series
Let f(x) = ck(x – a)k for all x in an open interval
(a – R, a + R) for some R, then the series ck(x – a)k
is the Taylor series P(x) of f(x).
Σk=0
∞
Theorem (Uniqueness theorem for Taylor series) :
The Taylor series of f(x) is the only power series that could be the same as f(x).