1

Appendix E: Sigma Notation

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Definition: Sequence

• A sequence is a function a(n) (written an) who’s domain is the set of natural numbers {1, 2, 3, 4, 5, ….}. an is called the general term of the sequence.

• The output of a sequence can be written as {a1, a2, a3, …, an-1, an, an+1, …}, where an is a term in a sequence, an-1 is the term before it, and an+1 is the term after it.

• Sequences can be either finite (their domains are {1, 2, 3, …, n}) or infinite (their domains are {1, 2, 3, ….}).

• A sequence who’s input for the next term in the sequence is the value of the previous term is called a recursive sequence.

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Definition: Arithmetic Sequence

An arithmetic sequence is a sequence generated by adding a real number (called the common difference, d) to the previous term to get the next term. The general term of an arithmetic is given by an = a1 + d(n – 1) where a1 and d are any real numbers.

Example Find the general term of the 7/3, 8/3, 3, 10/3, ….

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Definition: Geometric Sequence

A geometric sequence is a sequence generated by multiplying the previous term by a real number (called the common ratio r). The general term of a geometric sequence is given by an = a1 r(n – 1) where a1 and r are any real numbers, is called an geometric sequence.

Example Find the general term sequence 2, 2/5, 2/25, 2/125, …

TI: seq(ax , x, i start, i stop)

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Definition: Series

• A finite series is the sum of a finite number of terms of a sequence.

• An infinite series is the sum of an infinite number of terms of a sequence.

• We use sigma notation to denote a series. The series does not have to start at i = 1, but i must be in the domain of ai.

1 21

...n

i ni

a a a a

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Definition: Geometric Sequence

The nth partial sum is the sum of the first n terms of a sequence. It MUST start at i = 1 with partial sum notation.

1 21

...n

n i ni

S a a a a

An infinite sum is the sum of all the terms of an infinite sequence.

1 21

...ii

S a a a

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Definition: Example

.1

2Evaluate 1.

7

2 i i

5

0

3. Evaluate .k

k

x

43

2

2. Evaluate 2 .k

k

TI: sum(seq(ax , x, i start, i stop))

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Definition: Example

4. Write the sum in sigma notation.

3 4 5 6 7

5 7 9 11 13

5. Write the sum in sigma notation.

1+3+5+7+...+(2 -1)n

2 3

6. Write the sum in sigma notation.

1 ... ( 1)n nx x x x

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Definition: Series

For a finite arithmetic series,

For an infinite arithmetic series,

For a finite geometric series,

For an infinite geometric series, if | r | < 1. It DNE otherwise.

12n n

nS a a

S DNE

1 1

1

n

n

a rS

r

1

1

aS

r

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Definition: Example

7. Find the 97th partial sum of 2 3.na n

1

18. Evaluate 2 .

3

i

i

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Definition: Series Formulas

1 1

1 1 1

.

.

n n

i ii i

n n n

i i i ii i i

a ca c a

b a b a b

Let c be a constant and n a positive integer.

1

1

. 1

.

n

i

n

i

c n

d c cn

2

1

3 22

1

.2 2

.3 2 6

n

i

n

i

n ne i

n n nf i

2

3

1

1.

2

n

i

n ng i

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Definition: Series Formulas

2

1

3n

i

i

9. Write a formula for the series in terms of n:

10. If the interval [a, b] is split into n equal subintervals, write a sequence xi that represents the x coordinate of the left side, midpoint, and right side of each subinterval.

11. Show that 1

1 3lim 1

2

n

ni

i

n n