One of the definitions of ee involves the infinite series 1 + 1 + 1 / 2 + 1 / 6 + 1 / 24 + ⋯ + 1 / n! + ⋯. a generalization exists to
define eexx:
e^x = 1 + x +x^2 / 2 +x3 // 6 + x4 /24 + ⋯ + x6n / n! + ⋯
this series definition of eexx allows us to approximate powers of the transcendental number ee using strictly rational
numbers. this definition is accurate for all real numbers.
a. verify that the formula given for ee can be obtained by plugging xx = 1 into the formula for eexx.
b. use the first seven terms of the series to calculate ee, ee2, and ee3.
c. use the inverse of yy = eexx to see how accurate your answer to part (b) is.
d. newer calculators and computers use these types of series carried out to as many terms as needed to produce
their results for operations that are not otherwise obvious. it may seem cumbersome to calculate these by
hand knowing that computers can calculate hundreds and thousands of terms of these series in a single
second. use a calculator or computer to compare how accurate your results from part (b) were to the value
given by your technology.