1051 11.2 Arithmetic Sequences and Series 99. Approximating ln11 +x2 The series x - x2 2 + x3 3 - x4 4 + g can be used to approximate the value of ln 11 + x2 for values of x in 1-1, 14. Use the first six terms of this series to approximate each expression. Compare this approximation with the value obtained on a calculator. (a) ln 1.02 1x = 0.022 (b) ln 0.97 1x = -0.032 100. Approximating P Find the sum of the first six terms of the series p4 90 = 1 14 + 1 24 + 1 34 + 1 44 + 1 54 + g+ 1 n4 + g . Multiply this result by 90, and take the fourth root to obtain an approximation of p. Compare this answer to the actual decimal approximation of p. 101. Approximating Powers of e The series e a ≈1 + a + a2 2! + a3 3! + g+ an n! , where n! = 1 # 2 # 3 # 4 # g# n, can be used to approximate the value of e a for any real number a. Use the first eight terms of this series to approximate each expression. Compare this approximation with the value obtained on a calculator. (a) e (b) e-1 102. Approximating Square Roots The recursively defined sequence a1 = k an = 1 2 aan-1 + k an-1b, if n 71 can be used to compute 2k for any positive number k. This sequence was known to Sumerian mathematicians 4000 years ago, and it is still used today. Use this sequence to approximate the given square root by finding a6. Compare the result with the actual value. (Data from Heinz-Otto, P., Chaos and Fractals, Springer-Verlag.) (a) 22 (b) 211 11.2 Arithmetic Sequences and Series ■ Arithmetic Sequences ■ Arithmetic Series Arithmetic Sequences An arithmetic sequence (or arithmetic progression) is a sequence in which each term after the first differs from the preceding term by a fixed constant, called the common difference. The sequence 5, 9, 13, 17, 21, c is an arithmetic sequence because each term after the first is obtained by adding 4 to the previous term. That is, 9 = 5 + 4, 13 = 9 + 4, 17 = 13 + 4, 21 = 17 + 4, and so on. The common difference is 4. If the common difference of an arithmetic sequence is d, then d =an+1 −an, Common difference d for every positive integer n in the domain of the sequence.
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