Theorem 17 The Rearrangement Theorem For Absolutely Convergent Series If E 1an Converges Absolutely And B B 1 (149.55 KiB) Viewed 35 times
THEOREM 17-The Rearrangement Theorem for Absolutely Convergent Series If E-1an converges absolutely, and b₁, b₂, . . . , bµ‚ . . . is any arrangement of the sequence {a}, then Σb, converges absolutely and Σbn n=1 = n=1 an.
85. a. The series 1 - · 1/2 + 1 - 11 + 12/17 - 31/3 + · · · + 3 − 2 + 3 does not meet one of the conditions of Theorem 15. Which one? b. Use Theorem 17 to find the sum of the series in part (a).
THEOREM 15-The Alternating Series Test The series ∞ 3. Un Σ(-1)"+¹ un = U₁₁ = U₂+Uz - U₁ + n=1 converges if the following conditions are satisfied: 1. The un's are all positive. 2. The un's are eventually nonincreasing: uun+1 for all n ≥ N, for some integer N. 0.
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