Section 8.1. Sequences and Series

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Transcription:

Section 8.1 Sequences and Series

Sequences

Definition A sequence is a list of numbers.

Definition A sequence is a list of numbers. A sequence could be finite, such as: 1, 2, 3, 4

Definition A sequence is a list of numbers. A sequence could be finite, such as: 1, 2, 3, 4 A sequence can also be infinite, such as: 2, 4, 8, 16, 32, 64,... Here the... just means that the pattern continues forever.

Sequence Notation A series can also be thought of as a function where you only plug in natural numbers.

Sequence Notation A series can also be thought of as a function where you only plug in natural numbers. For example, 2, 4, 8, 16, 32, 64,... can be thought of as a(n) = 2 n since a(1) = 2 1 = 2, a(2) = 2 2 = 4, a(3) = 2 3 = 8, etc

Sequence Notation A series can also be thought of as a function where you only plug in natural numbers. For example, 2, 4, 8, 16, 32, 64,... can be thought of as a(n) = 2 n since a(1) = 2 1 = 2, a(2) = 2 2 = 4, a(3) = 2 3 = 8, etc Normally, however, the sequence is written with the input variable as a subscript: a n = 2 n Here a 1 = 2, a 2 = 4, a 3 = 8, etc

Examples 1. Find the first three terms of the sequence a n = n+1 n.

Examples 1. Find the first three terms of the sequence a n = n+1 n. a 1 = 2, a 2 = 3 2, a 3 = 4 3

Examples 1. Find the first three terms of the sequence a n = n+1 n. a 1 = 2, a 2 = 3 2, a 3 = 4 3 2. Find the first four terms of the sequence a n = ( 1) n n 3

Examples 1. Find the first three terms of the sequence a n = n+1 n. a 1 = 2, a 2 = 3 2, a 3 = 4 3 2. Find the first four terms of the sequence a n = ( 1) n n 3 a 1 = 1, a 2 = 8, a 3 = 27, a 4 = 64

Examples 1. Find the first three terms of the sequence a n = n+1 n. a 1 = 2, a 2 = 3 2, a 3 = 4 3 2. Find the first four terms of the sequence a n = ( 1) n n 3 a 1 = 1, a 2 = 8, a 3 = 27, a 4 = 64 3. Find the formula for the general term a n of the sequence 2, 4, 6, 8, 10,...

Examples 1. Find the first three terms of the sequence a n = n+1 n. a 1 = 2, a 2 = 3 2, a 3 = 4 3 2. Find the first four terms of the sequence a n = ( 1) n n 3 a 1 = 1, a 2 = 8, a 3 = 27, a 4 = 64 3. Find the formula for the general term a n of the sequence 2, 4, 6, 8, 10,... a n = 2n

Examples 1. Find the first three terms of the sequence a n = n+1 n. a 1 = 2, a 2 = 3 2, a 3 = 4 3 2. Find the first four terms of the sequence a n = ( 1) n n 3 a 1 = 1, a 2 = 8, a 3 = 27, a 4 = 64 3. Find the formula for the general term a n of the sequence 2, 4, 6, 8, 10,... a n = 2n 4. Find the formula for the general term a n of the sequence 2, 6, 18, 54, 162,...

Examples 1. Find the first three terms of the sequence a n = n+1 n. a 1 = 2, a 2 = 3 2, a 3 = 4 3 2. Find the first four terms of the sequence a n = ( 1) n n 3 a 1 = 1, a 2 = 8, a 3 = 27, a 4 = 64 3. Find the formula for the general term a n of the sequence 2, 4, 6, 8, 10,... a n = 2n 4. Find the formula for the general term a n of the sequence 2, 6, 18, 54, 162,... a n = ( 1) n 2 3 n 1

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms.

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms. The most famous example of this type of sequence is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,...

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms. The most famous example of this type of sequence is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,... For this sequence, the first two numbers are 0 and 1. Every number after that is calculated by adding the two previous numbers:

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms. The most famous example of this type of sequence is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,... For this sequence, the first two numbers are 0 and 1. Every number after that is calculated by adding the two previous numbers: a 3 = 0 + 1 = 1

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms. The most famous example of this type of sequence is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,... For this sequence, the first two numbers are 0 and 1. Every number after that is calculated by adding the two previous numbers: a 3 = 0 + 1 = 1 a 4 = 1 + 1 = 2

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms. The most famous example of this type of sequence is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,... For this sequence, the first two numbers are 0 and 1. Every number after that is calculated by adding the two previous numbers: a 3 = 0 + 1 = 1 a 4 = 1 + 1 = 2 a 5 = 1 + 2 = 3

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms. The most famous example of this type of sequence is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,... For this sequence, the first two numbers are 0 and 1. Every number after that is calculated by adding the two previous numbers: a 3 = 0 + 1 = 1 a 4 = 1 + 1 = 2 a 5 = 1 + 2 = 3 a 6 = 2 + 3 = 5

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms. The most famous example of this type of sequence is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,... For this sequence, the first two numbers are 0 and 1. Every number after that is calculated by adding the two previous numbers: a 3 = 0 + 1 = 1 a 4 = 1 + 1 = 2 a 5 = 1 + 2 = 3 a 6 = 2 + 3 = 5 a 7 = 3 + 5 = 8

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms. The most famous example of this type of sequence is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,... For this sequence, the first two numbers are 0 and 1. Every number after that is calculated by adding the two previous numbers: a 3 = 0 + 1 = 1 a 4 = 1 + 1 = 2 a 5 = 1 + 2 = 3 a 6 = 2 + 3 = 5 a 7 = 3 + 5 = 8 a 8 = 5 + 8 = 13

Recursive Sequences Some sequences can be written recursively, i.e. the terms are calculated from previous terms. The most famous example of this type of sequence is the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,... For this sequence, the first two numbers are 0 and 1. Every number after that is calculated by adding the two previous numbers: a 3 = 0 + 1 = 1 a 4 = 1 + 1 = 2 a 5 = 1 + 2 = 3 a 6 = 2 + 3 = 5 a 7 = 3 + 5 = 8 a 8 = 5 + 8 = 13 a 9 = 8 + 13 = 21

Examples Find the first five terms of each sequence. 1. a n+1 = a n, a 1 = 256

Examples Find the first five terms of each sequence. 1. a n+1 = a n, a 1 = 256 a 1 = 256, a 2 = 16, a 3 = 4, a 4 = 2, a 5 = 2

Examples Find the first five terms of each sequence. 1. a n+1 = a n, a 1 = 256 a 1 = 256, a 2 = 16, a 3 = 4, a 4 = 2, a 5 = 2 2. a n+1 = a n a n 1, a 1 = 10, a 2 = 8

Examples Find the first five terms of each sequence. 1. a n+1 = a n, a 1 = 256 a 1 = 256, a 2 = 16, a 3 = 4, a 4 = 2, a 5 = 2 2. a n+1 = a n a n 1, a 1 = 10, a 2 = 8 a 1 = 10, a 2 = 8, a 3 = 18, a 4 = 10, a 5 = 8

Series and Sums

Definition Suppose you have an infinite sequence a n. The nth partial sum is the sum of the first n terms: S n = a 1 + a 2 +... + a n For example, if a n = n + 1, (a 1 = 2, a 2 = 3, a 3 = 4, a 4 = 5, a 5 = 6) S 5 = 2 + 3 + 4 + 5 + 6 = 20

Sigma Notation To represent sums, we use the uppercase Greek letter sigma,. For example: 4 (2k 1) = (2 1 1) + (2 2 1) + (2 3 1) + (2 4 1) k=1 = 1 + 3 + 5 + 7 = 16

Examples 1. Find the 6th partial sum of the sequence 5, 10, 15, 20,...

Examples 1. Find the 6th partial sum of the sequence 5, 10, 15, 20,... 105

Examples 1. Find the 6th partial sum of the sequence 5, 10, 15, 20,... 105 2. Evaluate the sum: 4 j=1 3 j + 1

Examples 1. Find the 6th partial sum of the sequence 5, 10, 15, 20,... 105 2. Evaluate the sum: 4 j=1 3 j + 1 77 20

Examples 1. Find the 6th partial sum of the sequence 5, 10, 15, 20,... 105 2. Evaluate the sum: 4 j=1 3 j + 1 77 20 3. Write the sum in sigma notation: 3 2 + 4 4 5 8 + 6 16 7 32

Examples 1. Find the 6th partial sum of the sequence 5, 10, 15, 20,... 105 2. Evaluate the sum: 4 j=1 3 j + 1 77 20 3. Write the sum in sigma notation: 3 2 + 4 4 5 8 + 6 16 7 32 5 k=1 ( 1) k (k + 2) 2 k

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