Section 9.7 Infinite Series: “Maclaurin and Taylor Polynomials”
X34 computation techniques for maclaurin expansions
Transcript of X34 computation techniques for maclaurin expansions
Computation Techniques for Maclaurin Expansions
Computation Techniques for Maclaurin Expansions Direct computation of the Mac-series can be messyvia derivatives.
Computation Techniques for Maclaurin Expansions Direct computation of the Mac-series can be messyvia derivatives. In this section we show some of the algebraic techniques for computing the Mac-series.
Computation Techniques for Maclaurin Expansions Direct computation of the Mac-series can be messyvia derivatives. In this section we show some of the algebraic techniques for computing the Mac-series. Theorem: Let F(x) and G(x) be the Mac-series of f(x) and and g(x) respectively.
Computation Techniques for Maclaurin Expansions Direct computation of the Mac-series can be messyvia derivatives. In this section we show some of the algebraic techniques for computing the Mac-series. Theorem: Let F(x) and G(x) be the Mac-series of f(x) and and g(x) respectively. I. The Mac-expansions respect +, –, * , and /, that is, the Mac-series of f + g, f – g, f*g, and f/g are F + G, F – G, F*G, and F/G respectively.
Computation Techniques for Maclaurin Expansions Direct computation of the Mac-series can be messyvia derivatives. In this section we show some of the algebraic techniques for computing the Mac-series. Theorem: Let F(x) and G(x) be the Mac-series of f(x) and and g(x) respectively. I. The Mac-expansions respect +, –, * , and /, that is, the Mac-series of f + g, f – g, f*g, and f/g are F + G, F – G, F*G, and F/G respectively.
II. Mac-series respect composition of functions.This is particularly useful if g(x) is a polynomial in which case the Mac-series of f(g(x)) is F(g(x)).
Computation Techniques for Maclaurin Expansions Direct computation of the Mac-series can be messyvia derivatives. In this section we show some of the algebraic techniques for computing the Mac-series. Theorem: Let F(x) and G(x) be the Mac-series of f(x) and and g(x) respectively. I. The Mac-expansions respect +, –, * , and /, that is, the Mac-series of f + g, f – g, f*g, and f/g are F + G, F – G, F*G, and F/G respectively.
II. Mac-series respect composition of functions.This is particularly useful if g(x) is a polynomial in which case the Mac-series of f(g(x)) is F(g(x)).We list below the basic Mac-series that we will use in our examples .
Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P. Mac-series of polynomials are themselves.
II. For ex, its Σk=0 k! .
xk∞x + 2!1 + x2
+ .. ++ 3!x3
n! ..xn
=
Σk=0 (2k+1)!
(-1)kx2k+1∞x –
3!x3
+ 5!x5
+ .. =7!x7
– III. For sin(x), its
IV. For cos(x), its Σk=0 (2k)!
(-1)kx2k∞+
4!x4
6!x6
8!x8
+ 1 – – – .. =2!x2
V. For , its(1 – x ) 1
1 + x + x2 + x3 + x4 .. = Σk=0
∞xk
Computation Techniques for Maclaurin Expansions
Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P. Mac-series of polynomials are themselves.
II. For ex, its Σk=0 k! .
xk∞x + 2!1 + x2
+ .. ++ 3!x3
n! ..xn
=
Σk=0 (2k+1)!
(-1)kx2k+1∞x –
3!x3
+ 5!x5
+ .. =7!x7
– III. For sin(x), its
IV. For cos(x), its Σk=0 (2k)!
(-1)kx2k∞+
4!x4
6!x6
8!x8
+ 1 – – – .. =2!x2
V. For , its(1 – x ) 1
1 + x + x2 + x3 + x4 .. = Σk=0
∞xk
Computation Techniques for Maclaurin Expansions
Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P. Mac-series of polynomials are themselves.
II. For ex, its Σk=0 k! .
xk∞x + 2!1 + x2
+ .. ++ 3!x3
n! ..xn
=
Σk=0 (2k+1)!
(-1)kx2k+1∞x –
3!x3
+ 5!x5
+ .. =7!x7
– III. For sin(x), its
IV. For cos(x), its Σk=0 (2k)!
(-1)kx2k∞+
4!x4
6!x6
8!x8
+ 1 – – – .. =2!x2
V. For , its(1 – x ) 1
1 + x + x2 + x3 + x4 .. = Σk=0
∞xk
Computation Techniques for Maclaurin Expansions
Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P. Mac-series of polynomials are themselves.
II. For ex, its Σk=0 k! .
xk∞x + 2!1 + x2
+ .. ++ 3!x3
n! ..xn
=
Σk=0 (2k+1)!
(-1)kx2k+1∞x –
3!x3
+ 5!x5
+ .. =7!x7
– III. For sin(x), its
IV. For cos(x), its Σk=0 (2k)!
(-1)kx2k∞+
4!x4
6!x6
8!x8
+ 1 – – – .. =2!x2
V. For , its(1 – x ) 1
1 + x + x2 + x3 + x4 .. = Σk=0
∞xk
Computation Techniques for Maclaurin Expansions
Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P. Mac-series of polynomials are themselves.
II. For ex, its Σk=0 k! .
xk∞x + 2!1 + x2
+ .. ++ 3!x3
n! ..xn
=
Σk=0 (2k+1)!
(-1)kx2k+1∞x –
3!x3
+ 5!x5
+ .. =7!x7
– III. For sin(x), its
IV. For cos(x), its Σk=0 (2k)!
(-1)kx2k∞+
4!x4
6!x6
8!x8
+ 1 – – – .. =2!x2
V. For , its(1 – x ) 1
1 + x + x2 + x3 + x4 .. = Σk=0
∞xk
Computation Techniques for Maclaurin Expansions
Summary of the Mac-series
I. For polynomials P, Mac-poly of degree k consists the first k-terms of the polynomial P. Mac-series of polynomials are themselves.
II. For ex, its Σk=0 k! .
xk∞x + 2!1 + x2
+ .. ++ 3!x3
n! ..xn
=
Σk=0 (2k+1)!
(-1)kx2k+1∞x –
3!x3
+ 5!x5
+ .. =7!x7
– III. For sin(x), its
IV. For cos(x), its Σk=0 (2k)!
(-1)kx2k∞+
4!x4
6!x6
8!x8
+ 1 – – – .. =2!x2
V. For , its(1 – x ) 1
1 + x + x2 + x3 + x4 .. = Σk=0
∞xk
Computation Techniques for Maclaurin Expansions
Example: Find the Mac-series of sin(x) + cos(x)
Computation Techniques for Maclaurin Expansions
Example: Find the Mac-series of sin(x) + cos(x)
Σk=0 (2k+1)!
(-1)kx2k+1∞
x – 3!x3
+5!x5
+ .. =7!x7
– sin(x) =
Computation Techniques for Maclaurin Expansions
Example: Find the Mac-series of sin(x) + cos(x)
Σk=0 (2k+1)!
(-1)kx2k+1∞
x – 3!x3
+5!x5
+ .. =7!x7
– sin(x) =
cos(x) = Σk=0 (2k)!
(-1)kx2k∞
+ 4!x4
6!x6
8!x8
+ 1 – – – .. =2!x2
Computation Techniques for Maclaurin Expansions
Example: Find the Mac-series of sin(x) + cos(x)
Σk=0 (2k+1)!
(-1)kx2k+1∞
x – 3!x3
+5!x5
+ .. =7!x7
– sin(x) =
cos(x) = Σk=0 (2k)!
(-1)kx2k∞
+ 4!x4
6!x6
8!x8
+ 1 – – – .. =2!x2
Computation Techniques for Maclaurin Expansions
Therefore,
sin(x) + cos(x) =1 + x – 2!x2
– 3!x3
+ 4!x4
+5!x5
6!x6
– 7!x7
– ..
Example: Find the Mac-series of sin(x) + cos(x)
Σk=0 (2k+1)!
(-1)kx2k+1∞
x – 3!x3
+5!x5
+ .. =7!x7
– sin(x) =
cos(x) = Σk=0 (2k)!
(-1)kx2k∞
+ 4!x4
6!x6
8!x8
+ 1 – – – .. =2!x2
Computation Techniques for Maclaurin Expansions
Therefore,
sin(x) + cos(x) =1 + x – 2!x2
– 3!x3
+ 4!x4
+5!x5
6!x6
– 7!x7
– ..
= Σk=0 (2k+1)!
(-1)kx2k+1∞
+(2k)!
(-1)kx2k
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2 Σ
k=0 k!xk∞
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2 Σ
k=0 k! xk+2∞
=Σk=0 k!
xk∞
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2 Σ
k=0 k! xk+2∞
=Σk=0 k!
xk∞
x +2!
1 + x2
+ ..+3!x3
) = x2(
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2
+2!
x2 + x3 x4
+ ..+3!x5
Σk=0 k!
xk+2∞=Σ
k=0 k!xk∞
=x +2!
1 + x2
+ ..+3!x3
) = x2(
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2
+2!
x2 + x3 x4
+ ..+3!x5
Σk=0 k!
xk+2∞=
Example: Find the Mac-series of sin(x2)
Σk=0 k!
xk∞
=x +2!
1 + x2
+ ..+3!x3
) = x2(
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2
+2!
x2 + x3 x4
+ ..+3!x5
Σk=0 k!
xk+2∞=
Example: Find the Mac-series of sin(x2)
Σk=0 (2k+1)!
(-1)kx2k+1∞
x – 3!x3
+5!x5
+ .. =7!x7
– sin(x) =
Σk=0 k!
xk∞
=x +2!
1 + x2
+ ..+3!x3
) = x2(
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2
+2!
x2 + x3 x4
+ ..+3!x5
Σk=0 k!
xk+2∞=
Example: Find the Mac-series of sin(x2)
Σk=0 (2k+1)!
(-1)kx2k+1∞
x – 3!x3
+5!x5
+ .. =7!x7
– sin(x) =
sin(x2) = Σk=0 (2k+1)!
(-1)k(x2)2k+1∞
Σk=0 k!
xk∞
=x +2!
1 + x2
+ ..+3!x3
) = x2(
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2
+2!
x2 + x3 x4
+ ..+3!x5
Σk=0 k!
xk+2∞=
Example: Find the Mac-series of sin(x2)
Σk=0 (2k+1)!
(-1)kx2k+1∞
x – 3!x3
+5!x5
+ .. =7!x7
– sin(x) =
sin(x2) = Σk=0 (2k+1)!
(-1)k(x2)2k+1∞
= Σk=0 (2k+1)!
(-1)kx4k+2∞
Σk=0 k!
xk∞
=x +2!
1 + x2
+ ..+3!x3
) = x2(
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2
+2!
x2 + x3 x4
+ ..+3!x5
Σk=0 k!
xk+2∞=
Example: Find the Mac-series of sin(x2)
Σk=0 (2k+1)!
(-1)kx2k+1∞
x – 3!x3
+5!x5
+ .. =7!x7
– sin(x) =
sin(x2) =
=
Σk=0 (2k+1)!
(-1)k(x2)2k+1∞
= Σk=0 (2k+1)!
(-1)kx4k+2∞
Σk=0 k!
xk∞
=
x2 – 3!
(x2)3+
5!(x2)5
+ .. = 7!
(x2)7–
x +2!
1 + x2
+ ..+3!x3
) = x2(
Example: Find the Mac-series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σk=0 k! .
xk∞x +
2!1 + x2
+ .. ++3!x3
n! xn
=+ ..
Therefore, x2ex = x2
+2!
x2 + x3 x4
+ ..+3!x5
Σk=0 k!
xk+2∞=
Example: Find the Mac-series of sin(x2)
Σk=0 (2k+1)!
(-1)kx2k+1∞
x – 3!x3
+5!x5
+ .. =7!x7
– sin(x) =
sin(x2) =
=
Σk=0 (2k+1)!
(-1)k(x2)2k+1∞
= Σk=0 (2k+1)!
(-1)kx4k+2∞
Σk=0 k!
xk∞
=
x2 – 3!
(x2)3+
5!(x2)5
+ .. = 7!
(x2)7– x2 –
3!x6
+5!x10
7!x14
– ..
x +2!
1 + x2
+ ..+3!x3
) = x2(
Example: Find the Mac-series of
Computation Techniques for Maclaurin Expansions
1 + x2 x
Example: Find the Mac-series of
Computation Techniques for Maclaurin Expansions
1 + x2 x
1 + x2 x = x
1 + x2 1
*
Example: Find the Mac-series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σk=0
xk∞
=
1 + x2 x
1 + x2 x = x
1 + x2 1
*
1 – x 1
Example: Find the Mac-series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σk=0
xk∞
=
1 + x2 x
1 + x2 x = x
1 + x2 1
*
1 – x 1
by writing 1 + x2
1 as 1 – (-x2)
1
Example: Find the Mac-series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σk=0
xk∞
=
1 + x2 x
1 + x2 x = x
1 + x2 1
*
1 – x 1
by writing 1 + x2
1 as 1 – (-x2)
1 with substitution, we get
1 + x2 1 Σ
k=0(-x2)k
∞
= =
Example: Find the Mac-series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σk=0
xk∞
=
1 + x2 x
1 + x2 x = x
1 + x2 1
*
1 – x 1
by writing 1 + x2
1 as 1 – (-x2)
1 with substitution, we get
1 + x2 1 Σ
k=0(-x2)k
∞
= Σk=0
(-1)kx2k∞
=
Example: Find the Mac-series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σk=0
xk∞
=
1 + x2 x
1 + x2 x = x
1 + x2 1
*
1 – x 1
by writing 1 + x2
1 as 1 – (-x2)
1 with substitution, we get
1 + x2 1
= 1 – x2 + x4 – x6 + x8 – x10 ..
Σk=0
(-x2)k∞
= Σk=0
(-1)kx2k∞
=
Example: Find the Mac-series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σk=0
xk∞
=
1 + x2 x
1 + x2 x = x
1 + x2 1
*
1 – x 1
by writing 1 + x2
1 as 1 – (-x2)
1 with substitution, we get
1 + x2 1
= 1 – x2 + x4 – x6 + x8 – x10 ..
Σk=0
(-x2)k∞
= Σk=0
(-1)kx2k∞
=
Therefore 1 + x2
x = x * Σk=0
(-1)kx2k∞
Example: Find the Mac-series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σk=0
xk∞
=
1 + x2 x
1 + x2 x = x
1 + x2 1
*
1 – x 1
by writing 1 + x2
1 as 1 – (-x2)
1 with substitution, we get
1 + x2 1
= 1 – x2 + x4 – x6 + x8 – x10 ..
Σk=0
(-x2)k∞
= Σk=0
(-1)kx2k∞
=
Therefore 1 + x2
x = x * Σk=0
(-1)kx2k∞
Σk=0
(-1)kx2k+1=