Unformatted text preview: as large as we desire. EXAMPLE 10.20 Show that ∞ X n =1 1 n diverges. Here the theorem does not apply: lim n →∞ 1 /n = 0, so it looks like perhaps the series converges. Indeed, if you have the fortitude (or the software) to add up the ﬁrst 1000 terms you will ﬁnd that 1000 X n =1 1 n ≈ 7 . 49 , so it might be reasonable to speculate that the series converges to something in the neighborhood of 10. But in fact the partial sums do go to inﬁnity; they just get big very, very slowly. Consider the following: 1 + 1 2 + 1 3 + 1 4 > 1 + 1 2 + 1 4 + 1 4 = 1 + 1 2 + 1 2...
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 Spring '07
 JonathanRogawski
 Math, Calculus, lim, Summation, Limit of a sequence

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