I. The document discusses techniques for computing Maclaurin series expansions of functions algebraically rather than via direct computation of derivatives. It presents theorems stating that Maclaurin series respect basic algebraic operations like addition, subtraction, multiplication, and division. It also states they respect function composition. II. Examples are provided of computing the Maclaurin series for sin(x)+cos(x), x^2*e^x, sin(x^2), and 1/(1+x^2). Basic series for exponential, sine, and cosine functions are summarized that are used in the examples.

1. Computation Techniques for Maclaurin Expansions

2. Computation Techniques for Maclaurin Expansions
Direct computation of the Mac–series can be messy
via derivatives.

3. Computation Techniques for Maclaurin Expansions
Direct computation of the Mac–series can be messy
via derivatives. In this section we show some of the
algebraic techniques for computing the Mac–series.

4. Computation Techniques for Maclaurin Expansions
Direct computation of the Mac–series can be messy
via 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.

5. Computation Techniques for Maclaurin Expansions
Direct computation of the Mac–series can be messy
via 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.

6. Computation Techniques for Maclaurin Expansions
Direct computation of the Mac–series can be messy
via 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)).

7. Computation Techniques for Maclaurin Expansions
Direct computation of the Mac–series can be messy
via 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 .

8. Summary of the Mac–series
I. For polynomials P, the Mac–poly of degree k
consists of the first k–terms of the polynomial P.
Mac–series of polynomials are just the polynomials.
II. ex = 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. sin(x) =
IV. cos(x) = Σk=0 (2k)!
(–1)kx2k∞
+
4!
x4
6!
x6
8!
x8
+1 – – – .. =
2!
x2
V.
(1 – x )
1
1 + x + x2 + x3 + x4 .. = Σk=0
∞
xk
Computation Techniques for Maclaurin Expansions
=

9. Example A. Find the Mac–series of sin(x) + cos(x)
Computation Techniques for Maclaurin Expansions

10. Example A. Find the Mac–series of sin(x) + cos(x)
Computation Techniques for Maclaurin Expansions
Σk=0 (2k+1)!
(–1)kx2k+1
x –
3!
x3
+
5!
x5
+ .. =
7!
x7
–sin(x) =

11. Example A. Find the Mac–series of sin(x) + cos(x)
Computation Techniques for Maclaurin Expansions
Σk=0 (2k+1)!
(–1)kx2k+1
x –
3!
x3
+
5!
x5
+ .. =
7!
x7
–sin(x) =
cos(x) = Σ
(2k)!
(–1)kx2k
+
4!
x4
6!
x6
8!
x8
+1 – – – .. =
2!
x2
k=0

12. Example A. Find the Mac–series of sin(x) + cos(x)
Computation Techniques for Maclaurin Expansions
Therefore,
cos(x) + sin(x)
1 + x –
2!
x2
–
3!
x3
+
4!
x4
+
5!
x5
6!
x6
–
7!
x7
– ..
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
=

13. Example A. Find the Mac–series of sin(x) + cos(x)
Computation Techniques for Maclaurin Expansions
Therefore,
cos(x) + sin(x)
1 + x –
2!
x2
–
3!
x3
+
4!
x4
+
5!
x5
6!
x6
–
7!
x7
– ..
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
= Σk=0 (2k+1)!
(–1)kx2k+1
+(2k)!
(–1)kx2k
=

14. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions

15. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σ
k=0 k! .
xk
x +
2!
1 +
x2
+ .. ++
3!
x3
n!
xn
=+ ..

16. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σ
k=0 k! .
xk
x +
2!
1 +
x2
+ .. ++
3!
x3
n!
xn
=+ ..
Therefore,
x2ex = x2 Σk=0 k!
xk
x +
2!
1 +
x2
+ ..+
3!
x3
)= x2 (

17. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σ
k=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
=
Σk=0 k!
xk
=
x +
2!
1 +
x2
+ ..+
3!
x3
)= x2 (

18. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σ
k=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 C. Find the Mac–series of sin(x2)
Σk=0 k!
xk
=
x +
2!
1 +
x2
+ ..+
3!
x3
)= x2 (

19. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σ
k=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 C. Find the Mac–series of sin(x2)
Σ
k=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 (

20. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σ
k=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 C. Find the Mac–series of sin(x2)
Σ
k=0 (2k+1)!
(–1)kx2k+1
x –
3!
x3
+
5!
x5
+ .. =
7!
x7
–sin(x) =
sin(x2) =
Σk=0 k!
xk
=
x2
–
3!
(x2)3
+ 5!
(x2)5
+ ..
7!
(x2)7
–
x +
2!
1 +
x2
+ ..+
3!
x3
)= x2 (
, so

21. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σ
k=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 C. Find the Mac–series of sin(x2)
Σ
k=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
=
x2
–
3!
(x2)3
+ 5!
(x2)5
+ ..
7!
(x2)7
–
x +
2!
1 +
x2
+ ..+
3!
x3
)= x2 (
, so

22. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σ
k=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 C. Find the Mac–series of sin(x2)
Σ
k=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
=
x2
–
3!
(x2)3
+ 5!
(x2)5
+ ..
7!
(x2)7
–
x2
–
3!
x6
+
5!
x10
7!
x14
– ..
x +
2!
1 +
x2
+ ..+
3!
x3
)= x2 (
, so

23. Example B. Find the Mac–series of x2ex.
Computation Techniques for Maclaurin Expansions
ex = Σ
k=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 C. Find the Mac–series of sin(x2)
Σ
k=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
= Σ
k=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 (
, so

24. Example D. Find the Mac–series of
Computation Techniques for Maclaurin Expansions
1 + x2
x

25. Example D. Find the Mac–series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σ xk
=
1 + x2
x
1 – x
1
by writing
1 + x2
1 as
1 – (–x2)
1
k=0
,

26. Example D. Find the Mac–series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σ
k=0
xk
=
1 + x2
x
1 – x
1
by writing
1 + x2
1 as
1 – (–x2)
1
1 + x2
1 Σ(–x2)k
=
k=0
we have
,

27. Example D. Find the Mac–series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σ
k=0
xk
=
1 + x2
x
1 – x
1
by writing
1 + x2
1 as
1 – (–x2)
1
1 + x2
1 Σ(–x2)k
= Σ(–1)kx2k
= k=0
k=0
we have
,

28. Example D. Find the Mac–series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σ
k=0
xk
=
1 + x2
x
1 – x
1
by writing
1 + x2
1 as
1 – (–x2)
1
1 + x2
1
= 1 – x2 + x4 – x6 + x8 – x10 ..
Σ(–x2)k
= Σ(–1)kx2k
= k=0
k=0
we have
,

29. Example D. Find the Mac–series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σ
k=0
xk
=
1 + x2
x
1 + x2
1
1 – x
1
by writing
1 + x2
1 as
1 – (–x2)
1
1 + x2
1
= 1 – x2 + x4 – x6 + x8 – x10 ..
Σ(–x2)k
= Σ(–1)kx2k
=
Therefore 1 + x2
x = x *
k=0
k=0
we have
,

30. Example D. Find the Mac–series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σ
k=0
xk
=
1 + x2
x
= x
1 + x2
1
1 – x
1
by writing
1 + x2
1 as
1 – (–x2)
1
1 + x2
1
= 1 – x2 + x4 – x6 + x8 – x10 ..
Σ(–x2)k
= Σ
k=0
(–1)kx2k
=
Therefore 1 + x2
x = x *
Σ
k=0
(–1)kx2k
k=0
we have
*
,

31. Example D. Find the Mac–series of
Computation Techniques for Maclaurin Expansions
Since = 1 + x + x2 + .. xn + .. Σ
k=0
xk
=
1 + x2
x
= x
1 + x2
1
1 – x
1
by writing
1 + x2
1 as
1 – (–x2)
1
1 + x2
1
= 1 – x2 + x4 – x6 + x8 – x10 ..
Σ
k=0
(–x2)k
= Σ
k=0
(–1)kx2k
=
Therefore 1 + x2
x = x *
Σ
k=0
(–1)kx2k
Σ
k=0
(–1)kx2k+1
=
we have
*
,