2Fibonacci 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.
3An 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:
4The 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:
5Then find the limit as k increases without bound.
6The value of ex can be approximated using the following series known as the exponential series.
7Ex 1 Use the first five terms of the exponential series and a calculator to approximate the value of e0.65 to the nearest hundredth.
9The 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.
10Ex 2 Use the first five terms of the trig series to find the value of