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**Unformatted text preview: **this section with a theorem concerning geometric series.
7 To make this more palatable, it is usually accepted that 0.3 = 1
3 so that 0.9 = 3 0.3 = 3 1
3 = 1. Feel better? 570 Sequences and the Binomial Theorem Theorem 9.2. Geometric Series: Given the sequence ak = ark−1 for k ≥ 1, where |r| < 1,
∞ ark−1 = a + ar + ar2 + . . . =
k=1 If |r| ≥ 1, the sum a + ar + ar2 a
1−r + . . . is not deﬁned. The justiﬁcation of the result in Theorem 9.2 comes from taking the formula in Equation 9.2 for the
sum of the ﬁrst n terms of a geometric sequence and examining the formula as n → ∞. Assuming
|r| < 1 means −1 < r < 1, so rn → 0 as n → ∞. Hence as n → ∞,
n ark−1 = a
k=1 1 − rn
1−r → a
1−r As to what goes wrong when |r| ≥ 1, we leave that to Calculus as well, but will explore some cases
in the exercises. 9.2 Summation Notation 9.2.1 571 Exercises 1. Find the following sums.
9 (a) 5 (5g + 3) 4 2j (c) 5 12
(i + 1)
4 g =4 j =0 i=1 8 2 (g) n (e) (h) 100 (b)
k=3 1
k (d) (3k − 5)x k (f) k=0 n=1
3 (−1) j =1 n=1 (n + 1)!
n!
5!
j ! (5 − j )! 2. Rewrite the sum using summation notatio...

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