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Geometric and Binomial Series Representations and their Applications in Mathematics, Lecture notes of Pre-Calculus

The concept of geometric and binomial series representations, their formulas, and their applications in mathematics. It includes examples of finite and infinite geometric series, binomial expansions, and exponential series, along with their complex counterparts. The document also introduces Maclaurin and Taylor series as generalizations of the binomial series.

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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Mathematical Series
Mathematical series representations are very useful tools for describing images or for solving/approximating
the solutions to imaging problems. The may be used to “expand” a function into terms that are individual
monomial expressions (i.e., “powers”) of the coordinate
Geometric Series
Adjacent terms in a geometric series exhibit a constant ratio, e.g., if the scale factor for adjacent terms in
the series is t, the series has the form:
1+t+t2+t3+···=
X
n=0
tn
If |t|<1, this solution converges to a simple (and EASILY remembered) expression:
X
n=0
tn=1
1tif |t|<1
This series pops up frequently in science and it is useful to remember the solution. We may “turn the
problem around” by using a truncated series as an approximation for the ratio:
1
1t=
X
n=0
tn
=
N
X
n=0
tn
where Nis some maximum power in the series
Examples:
1.
1
0.9=1
10.1=(0.1)0+(0.1)1+(0.1)2+(0.1)3+···
=1+0.1+0.01 + 0.001 + ··· =1.11111···
This se ri es co nverge s fa ir ly quickl y an d becau se of the sma l l va lue of t; th is s eries m ay be t ru ncated
after a few terms and still obtain a fairly accurate value.
2.
1
0.25 =4
=1
10.75 =(0.75)0+(0.75)1+(0.75)2+(0.75)3+(0.75)4+(0.75)5+···
=(1+0.75 + 0.5625 + 0.421875 + 0.31640635) + ···
=3.05078125 + ···<4
Note that this series converges slowly because t is relatively “large;” the sum of a few terms is a poor
approximation to the end result.
1
pf3
pf4
pf5

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Mathematical Series

Mathematical series representations are very useful tools for describing images or for solving/approximating the solutions to imaging problems. The may be used to “expand” a function into terms that are individual monomial expressions (i.e., “powers”) of the coordinate

Geometric Series

Adjacent terms in a geometric series exhibit a constant ratio, e.g., if the scale factor for adjacent terms in the series is t, the series has the form:

1 + t + t^2 + t^3 + · · · =

X^ ∞

n=

tn

If |t| < 1 , this solution converges to a simple (and EASILY remembered) expression:

X^ ∞

n=

tn^ = 1 1 − t

if |t| < 1

This series pops up frequently in science and it is useful to remember the solution. We may “turn the problem around” by using a truncated series as an approximation for the ratio:

1 1 − t

X^ ∞

n=

tn^ ∼=

X^ N

n=

tn

where N is some maximum power in the series

Examples:

= (0.1)^0 + (0.1)^1 + (0.1)^2 + (0.1)^3 + · · ·

This series converges fairly quickly and because of the small value of t; this series may be truncated after a few terms and still obtain a fairly accurate value.

1

  1. 25

= (0.75)^0 + (0.75)^1 + (0.75)^2 + (0.75)^3 + (0.75)^4 + (0.75)^5 + · · ·

Note that this series converges slowly because t is relatively “large;” the sum of a few terms is a poor approximation to the end result.

Finite Geometric Series

The truncated geometric series also may be rewritten into a simple expression. Consider the finite series that inclues N + 1 terms:

X^ N

n=

tn^ = 1 + t + t^2 + t^3 + · · · + tN

X^ ∞

n=

tn^ −

X^ ∞

n=N +

tn

We may write this as the difference of two infinite geometric series:

X^ N

n=

tn^ =

1 + t + t^2 + t^3 + · · · tN^ + tN^ +1^ + · · ·

tN^ +1^ + tN+2^ + · · ·

X^ ∞

n=

tn^ −

X^ ∞

n=N+

tn

Now change the summation variable for the second infinite series from n to u ≡ n − (N + 1) =⇒ n = u + N + 1: (^) ∞ X

n=N +

tn^ =

X^ ∞

u=

tu+(N^ +1)^ =

X^ ∞

u=

tutN^ +1^ = tN+1^ ·

X^ ∞

u=

tu^ = tN^ +1^ · 1 1 − t

The expressions for the two series may now be combined:

X^ N

n=

tn^ =

X^ ∞

n=

tn^ −

X^ ∞

n=N

tn

μ 1 1 − t

− tN+1^ ·

1 − t X^ N

n=

tn^ =

1 − tN^ + 1 − t

if |t| < 1

This is also often shown in the form where the maximum power in the series is N − 1 so that there are N terms: NX− 1

n=

tn^ =

1 − tN 1 − t

if |t| < 1

Examples:

X^4

n=

(0.1)n^ = 1 + 0.1 + 0.01 + 0.001 + 0.0001 = 1. 1111

t = 0. 1 , N = 4 =⇒ 1 −^ t

N + 1 − t

=^1 −^ (0.1)

5 1 − (0.1)

X^3

n=

(0.75)n^ = 2. 7344

1 − tN+ 1 − t

=^1 −^ (0.75)

4 1 − (0.75)

  1. Cube root

(1 − x)

(^13) = 3

p (1 − x) = 1 +

3 ·^ (−x) +

3

2 ·^ (−x)

3

6 ·^ (−x)

3 ·^ x^ −^

9 ·^ x

2 +^5

81 ·^ x

∼= 1 +^1

3 ·^ x^ −^

9 ·^ x

2 ∼= 1 +^1

3 ·^ x

μ 15 16

μ 1 − 1 16

∼= 1 +^1

μ − 1 16

=^47

μ 1 −

∼= 1 +^1

μ −

μ 1 16

μ 1 −

∼= 1 +^1

μ −

μ −

μ −

Exponential Series

Without proof, we can write e raised to a numerical power u as a series in the powers of u:

exp [u] = eu^ =

X^ ∞

n=

un n!

=^1 0!

  • u 1!

  • u

2 2!

  • u

3 3!

= 1 + u +^1 2

u^2 +^1 6

u^3 + · · ·

which may be easily generalized to any base:

au^ = exp [u · log [a]] =

X^ ∞

n=

[u · log [a]]n n!

e.g.

10 u^ =

X^ ∞

n=

[u · log [10]]n n!

∼=

X^ ∞

n=

(log [10])n n!

· un

where log [10] ∼= 2. 302585

Complex Exponential Series

The generalization of the exponential series for complex-valued powers:

exp [±iθ] = e±iθ^ =

X^ ∞

n=

(±iθ)n n!

=^1 0!

± (iθ) 1!

  • (iθ)

2 2!

∓ (iθ)

3 3!

  • (iθ)

4 4!

= 1 ± iθ +^1 2

i^2 θ^2 ∓ 1 6

i^3 θ^3 +^1 24

i^4 θ^4 ± 1 120

i^5 θ^5 + · · ·

μ 1 −

θ^2 2

θ^4 24

± i ·

μ θ −

θ^3 6

θ^5 120

From Euler relation: exp [+iθ] = cos [θ] + i sin [θ]

Equate real and imaginary parts:

cos [θ] =

0! −^

θ^2 2! +^

θ^4 4! −^ · · ·^ = 1^ −^

θ^2 2 +^

θ^4 24 −^ · · · θ^ lim→ 0 cos [θ] = 1

sin [θ] = θ −

θ^3 3!

θ^5 5!

− · · · = θ −

θ^3 6

θ^5 120

lim θ→ 0

sin [θ] = θ