121616949-math.268

# 121616949-math.268 - sum ﬁnite You might guess from what...

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254 Chapter 10 Sequences and Series If X n =0 a n diverges, so does X n =0 b n . Exercises Determine whether the series converge or diverge. 1. X n =1 1 2 n 2 + 3 n + 5 2. X n =2 1 2 n 2 + 3 n - 5 3. X n =1 1 2 n 2 - 3 n - 5 4. X n =1 3 n + 4 2 n 2 + 3 n + 5 5. X n =1 3 n 2 + 4 2 n 2 + 3 n + 5 6. X n =1 ln n n 7. X n =1 ln n n 3 8. X n =2 1 ln n 9. X n =1 3 n 2 n + 5 n 10. X n =1 3 n 2 n + 3 n Roughly speaking there are two ways for a series to converge: As in the case of 1 /n 2 , the individual terms get small very quickly, so that the sum of all of them stays finite, or, as in the case of ( - 1) n - 1 /n , the terms don’t get small fast enough ( 1 /n diverges), but a mixture of positive and negative terms provides enough cancellation to keep the
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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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