By Kurtulus Oztopcu
Ebook by means of Oztopcu, Kurtulus
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Additional resources for Elementary Azerbaijani (Turk Dilleri Arastirmalari Dizisi)
947975 3 0 20 n FIGURE 2 ϱ EXAMPLE 3 Is the series ͚2 2n 3 1Ϫn convergent or divergent? n1 SOLUTION Let’s rewrite the nth term of the series in the form ar nϪ1: Another way to identify a and r is to write out the first few terms: ■ ■ 4ϩ 16 3 ϱ ͚2 ϩ 649 ϩ и и и 2n 3 1Ϫn n1 ϱ ͚ ͑2 ͒ 3 2 n Ϫ͑nϪ1͒ n1 ϱ ͚ n1 ϱ 4n 4 nϪ1 nϪ1 ͚ 4 ( 3 ) 3 n1 4 We recognize this series as a geometric series with a 4 and r 3 . Since r Ͼ 1, the series diverges by (4). 3171717. . as a ratio of integers. 3171717.
The remainder Tn for the comparison series 1͞n 3 was estimated in Example 6. 00005. 3 Exercises 1. Draw a picture to show that ϱ ͚ n2 9–10 Use the Comparison Test to determine whether the series is convergent or divergent. 3 ϱ 9. ͚ n1 What can you conclude about the series? ■ x ജ 1 and an f ͑n͒. By drawing a picture, rank the following three quantities in increasing order: 6 1 5 ͚a f ͑x͒ dx i1 11. 1 ϩ ϱ i 12. i2 ϱ bn is known to be convergent. 13. (a) If a n Ͼ bn for all n, what can you say about a n?
So, by taking the sum of sufficiently many terms, we can get as close as we like to the number 3. The table shows the first ten partial sums sn and the graph in Figure 2 shows how the sequence of partial sums approaches 3. 947975 3 0 20 n FIGURE 2 ϱ EXAMPLE 3 Is the series ͚2 2n 3 1Ϫn convergent or divergent? n1 SOLUTION Let’s rewrite the nth term of the series in the form ar nϪ1: Another way to identify a and r is to write out the first few terms: ■ ■ 4ϩ 16 3 ϱ ͚2 ϩ 649 ϩ и и и 2n 3 1Ϫn n1 ϱ ͚ ͑2 ͒ 3 2 n Ϫ͑nϪ1͒ n1 ϱ ͚ n1 ϱ 4n 4 nϪ1 nϪ1 ͚ 4 ( 3 ) 3 n1 4 We recognize this series as a geometric series with a 4 and r 3 .
Elementary Azerbaijani (Turk Dilleri Arastirmalari Dizisi) by Kurtulus Oztopcu