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12.7 (Chapter 9) Special Sequences & Series. Fibonacci Sequence: 1, 1, 3, 5, 8, 13, … Describes many patterns of numbers found in nature. a 1 = 1 and.

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Presentation on theme: "12.7 (Chapter 9) Special Sequences & Series. Fibonacci Sequence: 1, 1, 3, 5, 8, 13, … Describes many patterns of numbers found in nature. a 1 = 1 and."— Presentation transcript:

1 12.7 (Chapter 9) Special Sequences & Series

2 Fibonacci Sequence: 1, 1, 3, 5, 8, 13, … Describes many patterns of numbers found in nature. a 1 = 1 and a 2 = 1 How do we arrive at the next term? It was used to investigate the reproductive habits of rabbits in ideal conditions in 1202.

3 An important series used to define the irrational number e, developed by Leonhard Euler. It can be expressed as the sum of the following infinite series:

4 The binomial theorem can be used to derive the series for e. Let k be any positive integer and apply the binomial theorem to:

5 Then find the limit as k increases without bound.

6 The value of e x can be approximated using the following series known as the exponential series.

7 Ex 1 Use the first five terms of the exponential series and a calculator to approximate the value of e 0.65 to the nearest hundredth.

8 Trigonometric Series

9 The two trig series are convergent for all values of x. By replacing x with any angle measure expressed in radians and carrying out the computations, approximate values of the trig functions can be found to any desired degree of accuracy.

10 Ex 2 Use the first five terms of the trig series to find the value of

11 Eulers Formula

12 Therefore:

13 Ex 3 Write in exponential form:

14 Recall: There is no real number that is the logarithm of a negative number. You can use a special case of Eulers Formula to find a complex number that is the natural logarithm of a negative number.

15

16 Ex 4 Evaluate: ln(-540) ln(-270)


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