2 Fibonacci Sequence: 1, 1, 3, 5, 8, 13, …Describes many patterns of numbers found in nature.a1 = 1 and a2 = 1How 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 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 ex 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 e0.65 to the nearest hundredth.
9 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.
10 Ex 2 Use the first five terms of the trig series to find the value of
14 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.