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HW5Notes

HW5Notes - Notes on Homework#5 Exercise 2.7.1 c We are...

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Notes on Homework #5 Exercise 2.7.1 c We are going to prove the Alternating Series Test. The hypotheses are that ( a n ) is a sequence with a n 0, a 1 a 2 ≥ · · · a n a n +1 ≥ · · · , and ( a n ) 0. 1) The important thing to recognize is that the even-numbered partial sums end with a subtraction and the odd-numbered ones end with an addition: s 2 n = a 1 - a 2 + · · · - a 2 n , s 2 n +1 = a 1 - a 2 + · · · - a 2 n + a 2 n +1 . So s 2 n +2 - s 2 n = a 2 n +1 - a 2 n +2 , which is nonnegative since a 2 n +1 a 2 n +2 . This shows that ( s 2 n ) is a monotone increasing sequence. For the odd-numbered partial sums, s 2 n +1 - s 2 n - 1 = - a 2 n + a 2 n +1 0 because a 2 n +1 a 2 n , so ( s 2 n +1 ) is monotone decreasing. Finally, since s 2 n +1 = s 2 n + a 2 n +1 and a 2 n +1 0, we see that s 2 n s 2 n +1 . This verifies the inequalities in the hint: s 2 s 4 ≤ · · · ≤ s 2 n ≤ · · · ≤ s 2 n +1 ≤ · · · ≤ s 3 s 1 .

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HW5Notes - Notes on Homework#5 Exercise 2.7.1 c We are...

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