I. Motivation for studying infinite sequences and
series:
If every function f(x) were a polynomial,
calculus would be a piece of cake. It would be easy to differentiate
and to integrate and to find any particular value f(x_{0}).
Unfortunately, not even every differentiable function whose domain is the whole real line is a polynomial. For example f(x) = e^{x} couldn't be a polynomial because e^{x} --> + as x --> while on the other hand, e^{x} --> 0 as x --> -. This is not true for any polynomial P(x). Polynomials of degree 0 (constant functions P(x) = c) have both limits equal to c, while other polynomials have both limits equal to ±.
The desire to have useful polynomial-like
formulas for functions was a prime motivation for Isaac Newton in the
1660's when he first studied the calculus and created much of what we
think of as modern calculus. Two thousand years earlier the
foundational ideas of calculus were first broached when the ancient
Greeks asked difficult and intriguing questions about the infinite
division of time and space and discovered that there were irrational
numbers like (2)
These strains in our intellectual heritage lead to the following two
main lines of motivation for the subject of infinite sequences and
series in Math 112:
1) The mysteries of the ``infinitely small'' and
``infinitely large'':
-- What happens when we add up infinitely many smaller and vanishingly smaller numbers, as in Zeno's paradoxes (promulgated by Zeno in the fifth century B.C.; see page 51 of the text)2) The desire for polynomial formulas for functions --- even if we are forced to let the ``polynomials'' be infinitely long.
-- Is .99999999... = 1.00000000... ?
The result we come to in the end will be what is called the MacLaurin Series (or Taylor Series) formula for a function. For example, we shall write:
e^{x} | = | 1 + x + x^{2}/2! + x^{3}/3! + ... + x^{n}/n! + ... (forever) |
cos x | = | 1 - x^{2}/2! + x^{4}/4! - x^{6}/6! + ... |
sin x | = | x -
x^{3}/3! +
x^{5}/5! -
x^{7}/7! + ... (where n!= (1)(2)(3)...(n)) |
a) d/dx( e^{x} ) = e^{x} | b) d/dx( sin x ) = cos x |