32 computation techniques for maclaurin expansions
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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 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 Σk=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 Σk=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 + x3x4
+ ..+3!x5
Σk=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 + x3x4
+ ..+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 + x3x4
+ ..+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 + x3x4
+ ..+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 + x3x4
+ ..+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 + x3x4
+ ..+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 + x3x4
+ ..+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=