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Advanced Calculus by L. Loomis, S. Sternberg

By L. Loomis, S. Sternberg

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Example text

4. FINDING LIMITS 47 (b) Show by induction12 that for all k > 1, 1 < xk < 1. 2 (c) Show that this sequence is strictly increasing. (Hint: Try a few terms, maybe rewrite the formula, or think geometrically! ) (d) Show that this sequence converges. (e) You can probably guess what the limit appears to be. Can you prove that your guess is correct? 1 34. Show that the harmonic series ∞ k=1 k diverges to infinity, as follows (a version of this proof was given by Nicole Oresme (1323-1382) in 1350 [20, p.

This behavior is a form of divergence, but has some regularity that makes it a kind of “virtual convergence”. In the same way, the sequence of negative integers {−k}∞ k=1 = −1, −2, −3, . . diverges by marching to the “left end” of R, or −∞. It is common to use the convergence arrow in this “virtual” way. Strictly speaking, we shouldn’t, but if we’re going to sin, at least let us do it with flair8 . To be precise, 6 Note that (unlike the notation for limits of functions, later) we do not use a subscript like limk→∞ here, since there is no ambiguity about where the index of a sequence is going!

But for example by the time k = 20, both factors in xk are positive and increasing, so the sequence is eventually increasing . The third sequence xk = cos k is a bit harder to predict in the long run: but certainly for 0 ≤ k < π it decreases, while for π < k < 2π it increases, so already it is not monotone. We have seen that in general a sequence can be unbounded without necessarily diverging to infinity, or bounded without necessarily converging. However, a monotone sequence behaves much more predictably.

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