SECTION 12.4 The Limit of a Sequence; Infinite Series 889 Solution The terms of the sequence are … 1, 2, 1, 2, 1, 2, As n gets larger and larger, the terms continue to oscillate between 1 and 2, so the terms do not get closer to a single number L. The sequence is divergent. Examining a Divergent Sequence Discuss the sequence { } = ⎧ ⎨ ⎪ ⎩⎪⎪ s n n 1 if is odd 2 if is even n EXAMPLE 3 Determining Whether a Sequence Is Convergent or Divergent Determine whether the sequence { } ( ) { } = − s 1 2 n n converges or diverges. If it converges, find its limit. Solution EXAMPLE 4 The terms of the sequence are … − − − 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 64 , The terms alternate between being negative and being positive, but as n gets larger and larger, the terms get closer to 0. Figure 18 illustrates this.The sequence converges, and the limit of the sequence is 0. Figure 18 ( ) { } = − ⎧ ⎨ ⎪ ⎩⎪⎪ ⎫ ⎬ ⎪ ⎭⎪⎪ s 1 2 n n 6 x 4 2 y 20.5 0.5 Now Work PROBLEMS 9 AND 13 2 Define Infinite Series and Geometric Series Infinite Series Suppose you were asked to find the sum of an infinite collection of numbers. In particular, suppose you were asked to find the sum 1 2 1 4 1 8 1 16 . + + + + To help find the answer, consider the squares in Figure 19. Figure 19 1 unit 1 unit 1 4 – 1 2 – 1 8 – 1 16 –– The length of each side of the square in the figure is 1 unit, so the area of the square is 1 square unit.When we divide the square into two rectangles of equal area,
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