Sequences and Series Issues have come up in Physics involving a sequence or series of numbers being added or multiplied together. Sometimes we look at.

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Presentation transcript:

Sequences and Series Issues 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

How can we use a Recursive Definition?

More Examples

Arithmetic Sequences

Examples

Geometric Sequences

Examples 1)Which of the following sequences are geometric? 2, 4, 8, 16, ….; 1, 5, 9, 13, 17, ….; 2 3, 2 7, 2 11, )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 4 th and 5 th bounces?

Arithmetic and Geometric Means

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: – Infinite Sequence: 3, 7, 11, 15, … – Infinite Series: ….

Arithmetic Series

Summation Notation

Geometric Series

Infinite Geometric Series

Examples: