Overview
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Addition of real numbers is basically a binary operation: Given any two real numbers [math]a[/math] and [math]b[/math], there is defined a real number denoted by [math]a + b[/math] and called their sum. The sum of numbers [math]a_1, ..., a_n[/math], where [math]n \geq 3[/math], is then defined by repeated applications of the binary operation. For example, one way of grouping the terms is given by
[[math]]
( \cdots (((a_1 + a_2) + a_3) + a_4) + \cdots + a_{n-1}) + a_n .
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The Associative Law of Addition implies that the sums obtained by all the different possible groupings are the same; so we can discard the parentheses and write
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\sum_{i=1}^n a_i = a_1 + \cdots + a_n .
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Thus addition of any finite number of terms is defined. However, without further definitions, the sum of an infinite number of terns makes no sense at all. In this chapter we shall make the necessary definitions and develop the theory of infinite series. Two examples are
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\begin{eqnarray*}
\sum_{i = 0}^\infty \frac{1}{2^i} &=& 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots , \\
\sum_{i = 1}^\infty (-1)^i(2i + 1) &=& -3 + 5 - 7 + 9 - 11 + \cdots .
\end{eqnarray*}
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Later in the chapter we shall consider infinite series in which each term contains the power of an independent variable. An example is the series
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1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \cdots,
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which, for every real number [math]x[/math], is an infinite series of real nun bers. We shall see that many functions can be defined by these power series, and this fact is of fundamental importance in mathematics and its applications.
General references
Doyle, Peter G. (2008). "Crowell and Slesnick's Calculus with Analytic Geometry" (PDF). Retrieved Oct 29, 2024.