Download presentation

Presentation is loading. Please wait.

Published byLuke Nolan Modified over 2 years ago

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)

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google