892 CHAPTER 12 Sequences; Induction; the Binomial Theorem Geometric Series DEFINITION Geometric Series A special infinite series is the geometric series ∑ = + + + + + = ∞ − − ar a ar ar ar k k n 1 1 2 1 (2) where ≠ a 0 is the first term in the series and ≠ r 0 is the common ratio. To determine the conditions for convergence of the geometric series, examine the n th partial sum = + + + + − S a ar ar ar n n 2 1 (3) If = r 1, then a a a a a 1 k k 1 1 ∑ ⋅ = + + + + + = ∞ − and the n th partial sum is = + + + + = S a a a a na n In this case the sequence { } Sn diverges ( ( ) ≠ = =∞ →∞ →∞ a S na since 0, lim lim n n n if > a 0 and −∞ if < a 0), so the geometric series diverges. If = − r 1, the series becomes ∑ − = − + − + = ∞ − a a a a a ( 1) k k 1 1 and the n th partial sum is = ⎧ ⎨ ⎪ ⎩⎪⎪ S n a n 0 if is even if is odd n Since ≠ a 0, the sequence { } Sn diverges due to oscillating terms.Thus, the geometric series diverges for = − r 1. Suppose that ≠ r 1 and ≠ − r 1. To see what happens for other choices of r, ≠ r 0, multiply both sides of equation (3) by r, obtaining = + + + + rS ar ar ar ar n n 2 3 (4) Subtracting (4) from the expression for Sn in (3) gives ( ) ( ) ( ) ( ) − =++++ −++++ = − − = − − S rS a ar ar ar ar ar ar ar a ar S r a r 1 1 n n n n n n n 2 1 2 3 Factor both sides. Since ≠ r 1, express the n th partial sum of the geometric series as ( ) = − − = − − = − − − S a r r a ar r a r ar r 1 1 1 1 1 n n n n Now S a r ar r a r a r r a r a r r lim lim 1 1 lim 1 lim 1 1 1 lim n n n n n n n n n = − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − − ⋅ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − − ⋅ →∞ →∞ →∞ →∞ →∞ The limit of a difference equals the difference of the limits. The limit of a constant is the constant. The limit of the product of a constant and a function is the constant times the limit of the function. If − < < r 1 1, then = →∞ r lim 0, n n so = − →∞ S a r lim 1 . n n Thus, for − < < r 1 1, the geometric series converges, and its sum is − a r 1 .
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