Arithmetic Series james rahn算术级杰姆斯是.pptVIP

Section 9.2 If you start adding the terms of the arithmetic sequence 1, 2, 3, . . . , you get larger and larger values. Even if the terms of an arithmetic sequence are small, as in 0.001, 0.002, 0.003, . . . , the partial sums eventually get large. As the number of terms increases, the magnitude of the partial sum increases. But consider the geometric sequence 0.4, 0.04, 0.004, 0.0004, . . . . It has common ratio 1/10 , so the terms get smaller. The partial sums seem to follow a pattern. If you sum infinitely many terms of this sequence, would the result be infinitely large? It appears that the partial sums will not get infinitely large; they are all less than 0.5. The indicated sum of a geometric sequence is a geometric series. An infinite geometric series is a geometric series with infinitely many terms. In this lesson you will specifically look at convergent series, for which the sequence of partial sums approaches a long-run value as the number of terms increases. Jack baked a pie and promptly ate one-half of it. Determined to make the pie last, he then decided to eat only one-half of the pie that remained each day. Record the amount of pie eaten each day for the first seven days. For each of the seven days, record the total amount of pie eaten since it was baked. If Jack lives forever, then how much of this pie will he eat? The amount of pie eaten each day is a geometric sequence with first term 1/2 and common ratio 1/2. The first seven terms of this sequence are Find the partial sums, S1 through S7, of the terms in part a. It may seem that eating pie “forever” would result in eating a lot of pie. However, if you look at the pattern of the partial sums, it seems as though for any finite number of days Jack’s total is slightly less than 1. This leads to the conclusion that Jack would eat exactly one pie in the long run. This is a convergent infinite geometric series with long-run value 1. Recall that a geometric sequence can be represented with an explicit