Unformatted text preview: sum ﬁnite. You might guess from what we’ve seen that if the terms get small fast enough to do the job, then whether or not some terms are negative and some positive the series converges. THEOREM 10.36 If ∞ X n =0  a n  converges, then ∞ X n =0 a n converges. Proof. Note that 0 ≤ a n +  a n  ≤ 2  a n  so by the comparison test ∞ X n =0 ( a n +  a n  ) converges. Now ∞ X n =0 ( a n +  a n  )∞ X n =0  a n  = ∞ X n =0 a n +  a n    a n  = ∞ X n =0 a n converges by theorem 10.17 ....
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 Spring '07
 JonathanRogawski
 Math, Calculus, Sequences And Series, n=1, n=0

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