SECTION 12.3 Geometric Sequences; Geometric Series 879 Solution Figure 15(a) shows the result using a TI-84 Plus CE graphing calculator. Figure 15(b) shows the result using Desmos. The sum of the first 15 terms of the sequence 1 3 n { } ( ) is approximately 0.4999999652. (a) Now Work PROBLEM 47 4 Determine Whether a Geometric Series Converges or Diverges DEFINITION Infinite Geometric Series An infinite sum of the form a a r a r a rn 1 1 1 2 1 1 + + + + + − with first term a1 and common ratio r, is called an infinite geometric series and is denoted by a r k k 1 1 1 ∑ = ∞ − The sum Sn of the first n terms of a geometric series is S a a r a r a r n n 1 1 1 2 1 1 = + ⋅ + ⋅ + + ⋅ − (6) NOTE In calculus, limit notation is used, and the sum is written L S a r a r lim lim n n n k n k k k 1 1 1 1 1 1 ∑ ∑ = = = →∞ →∞ = − = ∞ − j If this finite sum Sn approaches a number L as n , →∞ then the infinite geometric series a r k k 1 1 1 ∑ = ∞ − converges to L and L is called the sum of the infinite geometric series . The sum is written as L a r k k 1 1 1 ∑= = ∞ − A series that does not converge is called a divergent series . THEOREM Convergence of an Infinite Geometric Series If r 1, < the infinite geometric series a r k k 1 1 1 ∑ = ∞ − converges. Its sum is a r a r 1 k k 1 1 1 1 ∑ = − = ∞ − (7) Intuitive Proof • If r 0, = then S a a 0 0 , n 1 1 = + + + = so formula (7) is true for r 0. = • If r 0 ≠ and r 1, < then, based on formula (3), S a r r a r a r r 1 1 1 1 n n n 1 1 1 = ⋅ − − = − − − (8) Since r 1, < it follows that rn approaches 0 as n . →∞ Then, in formula (8), the term a r r 1 n 1 − approaches 0, so the sum Sn approaches a r 1 1 − as n . →∞ ■ (b) Figure 15 NOTE A full proof is provided in the next section. j
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