890 CHAPTER 12 Sequences; Induction; the Binomial Theorem each rectangle has an area of 1 2 square unit. Now divide the yellow rectangle into two smaller rectangles. If this process is continued indefinitely, the original square of area 1 unit is subdivided into rectangles of area 1 2 , 1 4 , 1 8 , 1 16 , and so forth.Adding up the areas of the rectangles gives = + + + + = 1 1 2 1 4 1 8 1 16 area of square combined area of rectangles Surprised? Let’s look at this result from an algebraic point of view by starting with the infinite sum … + + + + 1 2 1 4 1 8 1 16 (1) Notice that (1) is the sum of a geometric series with 1st term a 1 2 1 = and common ratio r 1 2 . = As discussed in the previous section, its sum is a r 1 1 1. 1 1 2 1 2 − = − = Another way to look at the sum (1) is to use partial sums to see whether a trend develops. The first five partial sums are = + = + + = = + + + = = + + + + = = 1 2 0.5 1 2 1 4 0.75 1 2 1 4 1 8 7 8 0.875 1 2 1 4 1 8 1 16 15 16 0.9375 1 2 1 4 1 8 1 16 1 32 31 32 0.96875 First partial sum Second partial sum Third partial sum Fourth partial sum Fifth partial sum Each of these sums uses more of the terms in (1), and each sum seems to be getting closer to 1. DEFINITION Infinite Series If … … a a a , , , , n 1 2 is some infinite collection of numbers, then the expression ∑ = + + + + = ∞ a a a a k k n 1 1 2 is called an infinite series . The numbers … … a a a , , , , n 1 2 are called terms , and the number an is called the n th term or general term . The index of summation is not fixed for a given series. For example, the series in (1) may be written as ∑ + + + + + = = ∞ 1 2 1 2 1 2 1 2 1 2 n k k 2 3 1 This infinite series may also be written in any of the following equivalent forms: 1 2 1 2 1 2 1 2 k k k k k k 1 1 0 1 4 3 ∑ ∑ ∑ ( ) = ∞ − = ∞ + = ∞ − In most of our work, the index of summation will begin at 1.
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