Sequences and SeriesIssues have come up in Physics involving a sequence or series of numbers being added or multiplied together. Sometimes we look at an infinite series. Examples:

Book stack length as the number of books approaches infinity (requiring the CG to be supported by every lower book).

A bouncing ball which only reaches a fraction of the previous height on each succeeding bounce.

Terminology

• Sequence: Ordered list of numbers called terms: where is called the “nth” term.

• Explicit formula: gives the value of the nth term of a sequence in terms of . Example:Suppose Generate the first four terms of the sequence.

• Recursive Definition: Defines a sequence by specifying:– the first term of a series and – a recursive formula relating a term to the one before it.

How can we use a Recursive Definition?

• Examples:1)Generate the first 4 terms of this sequence.

2) Pyramid Building (page 566): Write a recursive definition for the following sequence: 1, 3, 6, 10, 15, 21, ....

3) For the same sequence, how can we find an explicit formula for the nth term? What is the 100th term?

More Examples1) The factorial of an integer (n) is expressed as n!

and means : (So 4! = 4 · 3 · 2 · 1 = 24.)Express the factorial function as a recursive definition.

2) Write a recursive definition for this sequence: 100, 10, 1, 0.1, 0.01, ....

Arithmetic Sequences• Sequence in which consecutive terms differ by the

same amount. They can be expressed as:

We refer to as the common difference.• How can we write a recursive definition for an

arithmetic sequence in terms of and

• How can we write an explicit formula for this sequence?

Examples1) Which of the following sequences are arithmetic?

a) 2, 4, 8, 16, ... b) 1, 5, 9, 13, 17, ...

2) What are the second and third terms of the following arithmetic sequence? 100, , , 82, ....

3) What is the 46th term of the arithmetic sequence that begins 3, 5, 7, ...?

4) For an arithmetic sequence, what is in terms of and ?

5) The number of seats in the first 13 rows in a stadium form an arithmetic sequence, with 14 seats in row 1 and 16 seats in row 2. How many seats are in row 13?

Geometric Sequences• Sequence such that every term forms the same

ratio () with its preceding term:

We refer to as the common ratio.

• How can we write a recursive definition for a geometric sequence?

• How can we write an explicit formula for ?

Examples1) Which of the following sequences are geometric?

2, 4, 8, 16, ….; 1, 5, 9, 13, 17, ….; 23, 27, 211, 215

2) What are the second and third terms of: 2, ?, ?, -54?

3) What is the tenth term of: 4, 12, 36, ….?4) The successive heights to which a bouncing ball

bounces form a geometric sequence. The ball rises to 100 cm on the first bounce, and to 49 cm on the third bounce. To what heights does it rise on the 4th and 5th bounces?

Arithmetic and Geometric Means• The arithmetic mean of two numbers x and y is

defined as : and is also called the average.In an arithmetic sequence a middle term is the arithmetic mean between its two surrounding terms.

• The geometric between two positive numbers x and y is defined as and is, by definition, positive.Suppose we have a geometric sequence: 3, ?, 12. What are the possible values of the middle term?One is the geometric mean; the other is its negative.

Series vs. Sequences

• A sequence of numbers is an ordered list of numbers called terms; the list can be finite or infinite.

• A series is what we get from adding the terms of a sequence. It, too, can be finite or infinite.

• Examples: – Finite sequence: 6, 9, 12, 15, 18– Finite Series: 6+9+12+15+18– Infinite Sequence: 3, 7, 11, 15, …– Infinite Series: 3+7+11+15+ ….

Arithmetic Series

• A series whose terms form an arithmetic sequence.• The sum of a finite arithmetic series:

is • Examples:

1) What is the sum of the even integers from 2 to 100?2) A salesman makes 10 sales on his first week, and

increases his weekly sales by 2 every following week. How many sales has he made at the end of 50 weeks?

Summation Notation• We use the Greek capital letter Sigma to indicate

summations, along with limits to designate the least and greatest values of used in the series.

• Examples: 1) How many terms are in this series?2) Express 7+ 11+ 15+ … + 203 + 207 in summation

notation.3) What is the sum of: ? Hint: When the expression inside

the Sigma is linear in , the series is arithmetic.4) What is the sum of: ?

Geometric Series• A geometric series is one whose terms form a

geometric sequence.• The sum of a finite geometric series:

is • Examples: Find the sums of:1) 3 + 6 + 12 + 24 + … + 3072.2) 3) -15 + 30 – 60 + 120 – 240 + 480.

Infinite Geometric Series

• An infinite geometric series with the first term: and a common ratio: has a finite sum:

We say that the series converges to the value S since the partial sums will get closer and closer to this final value.

• If the series does not have a finite sum; we say that the series diverges.

Examples:Do these infintie series converge? If so, to what?