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H* = 1, 0, 1, 0, 1, 0, K* = 3, 12, 39, 120, 363, 1092, 3279, L* = 1/2, 3/4, 7/8, 15/16, 31/32, The partial sums denoted by H* and K* don t settle down on anything But the partial sums denoted by L* seem to approach 1 They don t run away into uncharted territory, and they don t alternate between or among multiple numbers The partial sums in L* seem to have a clear destination that they could reach, if only they had an infinite amount of time to get there It turns out that the complete series L+, representing the sum of the infinite string of numbers in the sequence L, is exactly equal to 1! We can get an intuitive view of this fact by observing that the partial sums approach 1 As the position in the sequence of partial sums, L*, gets farther and farther along, the denominators keep doubling, and the numerator is always 1 less than the denominator In fact, if we want to find the nth number L*n in the sequence of partial sums L*, we can calculate it by using the following formula: L*n = (2n 1)/2n As n becomes large, 2n becomes large much faster, and the proportional difference between 2 1 and 2n becomes smaller When n reaches extremely large positive integer values, the quotient (2n 1)/2n is almost exactly equal to 1 We can make the quotient as close to 1 as we want by going out far enough in the series of partial sums, but we can never make it equal to or larger than 1 The sequence L* is said to converge on the number 1 The sequence of partial sums L* is an example of a convergent sequence The series L+ is an example of a convergent series
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Plotting a geometric sequence A geometric sequence, like an arithmetic sequence, appears as a set of points when plotted on a Cartesian plane Figure 19-2 shows examples of two geometric sequences as they appear
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Figure 19-2
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Rectangular-coordinate plots of two geometric sequences
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when graphed Note that the dashed curves, which show the general trends of the sequences (but aren t actually parts of the sequences), aren t straight lines, but they are smooth They don t turn corners or make sudden leaps One of the sequences in Fig 19-2 is increasing, and the dashed curve connecting this set of points goes upward as we move to the right Because this sequence is finite, the dashed curve ends at the point (6,32), where the term number is 6 and the term value is 32 The other sequence is decreasing, and the dashed curve goes downward and approaches 0 as we move to the right This sequence is infinite, as shown by the three dots at the end of the string of numbers, and also by the arrow at the right-hand end of the dashed curve If a geometric sequence has a negative factor, that is, if k < 0, the plot of the points alternates back and forth on either side of 0 The points fall along two different curves, one above the horizontal axis and the other below If you want to see what happens in a case like this, try plotting an example Set t 0 = 64 and k = 1/2, and plot the resulting points
An example Suppose you get a 5-year certificate of deposit (CD) at your local bank for \$100000, and it earns interest at the annualized rate of exactly 5 percent per year The CD will be worth \$127628 after 6 years To calculate this, multiply \$1000 by 105, then multiply this result by 105, and repeat this process a total of 5 times The resulting numbers form a geometric sequence:
After 1 year: \$100000 105 = \$105000 After 2 years: \$105000 105 = \$110250 After 3 years: \$110250 105 = \$115763 After 4 years: \$115763 105 = \$121551 After 5 years: \$121551 105 = \$127628
Another example Is the following sequence a geometric sequence If so, what are the values t0 (the starting value) and k (the factor of change)
T = 3, 6, 12, 24, 48, 96, This is a geometric sequence The numbers change by a factor of 2 In this case, t 0 = 3 and k = 2