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=