Unformatted text preview: fact, than positive term series. Consider pictorially what is going on in the alternating harmonic series, shown in ﬁgure 10.4 . Because the sizes of the terms a n are decreasing, the partial sums s 1 , s 3 , s 5 , and so on, form a decreasing sequence that is bounded below by s 2 , so this sequence must converge. Likewise, the partial sums s 2 , s 4 , s 6 , and so on, form an increasing sequence that is bounded above by s 1 , so this sequence also converges. Since all the even numbered partial sums are less than all the odd numbered ones, and...
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 Spring '07
 JonathanRogawski
 Math, Calculus, n=1

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