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**12.7 (Chapter 9) Special Sequences & Series**

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Fibonacci Sequence: 1, 1, 3, 5, 8, 13, … Describes many patterns of numbers found in nature. a1 = 1 and a2 = 1 How do we arrive at the next term? It was used to investigate the reproductive habits of rabbits in ideal conditions in 1202.

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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:

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**The binomial theorem can be used to derive the series for e**

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

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**Then find the limit as k increases without bound.**

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**The value of ex can be approximated using the following series known as the exponential series.**

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Ex 1 Use the first five terms of the exponential series and a calculator to approximate the value of e0.65 to the nearest hundredth.

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Trigonometric Series

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**The two trig series are convergent for all values of x**

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.

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

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Euler’s Formula

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Therefore:

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**Ex 3 Write in exponential form:**

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

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

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