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Berkeley MATH 1B - PDP Worksheet

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Rob Bayer Math 1B PDP Worksheet March 17, 2009Integral Test1. Determine whether each of the following series converge or diverge. DO NOT use the series comparison test.(a)∞Xn=13√nn(b) 1 +116+181+1256+ · · ·(c)∞Xn=1x29 + x6(d)∞Xn=31n(ln n)√ln ln n2. For which values of x does∞Xn=1(ln x)nconverge? And for∞Xn=1(ln n)x?3. Consider the series∞Xn=513√n − 3. Re-index this series to turn it into a p-series and determine if it converges ordiverges.If you finish either section early, move on to the “crazy facts” section below.Comparison Test1. Determine whether each of the following series converge or diverge.(a)∞Xn=1n − 3n3+ 4n + 2(b)∞Xn=4en2n− ln n(c)∞Xn=2sin2nn2+ ln n(d)∞Xn=11n + en(e)∞Xn=11n2n(f)∞Xn=2ln nn(g)∞Xn=3n2− 2n3+ 522. (a) Show that ifPanis a convergent series with non-negative terms, thenPa2nis also convergent.(b) However, ifPanis a convergent series with non-negative terms,P√ancould either converge or diverge.Give an example for each of these possibilities.(c) Show that ifPanandPbnare both convergent series with positive terms thenPanbnconverges too.Crazy Facts1. Find the flaw in the following “proof” that 0=1:0 = 0 + 0 + 0 + · · ·= (1 − 1) + (1 − 1) + (1 − 1) + · · ·= 1 − 1 + 1 − 1 + 1 − 1 + · · ·= 1 + (−1 + 1) + (−1 + 1) + · · ·= 1 + 0 + 0 + 0 + · · ·= 12. The Cantor Set is a set of real numbers constructed as follows: start with the interval [0, 1], and remove themiddle third of it. That is, remove the interval (13,23), leaving [0,13], [23, 1]. Now remove the middle third ofeach of these remaining intervals, leaving [0,19], [29,13], [23,79], [89, 1]. After continuing this process infinitely manytimes, you will be left with the Cantor set.(a) Show that the total length of all the intervals you remove is 1.(b) Convince yourselves that despite this, the cantor set has infinitely many numbers in it. Give someexamples of these numbers.(c) (Side note: it actually turns out that the Cantor Set is uncountable, meaning there are exactly the samenumber of numbers in it as there were in the interval [0, 1] before you started removing middle thirds. Theproof of this is actually very easy, but requires some knowledge of binary and ternary decimal systems.Talk to me if you’re curious.)3. Recall that the Fibonacci sequence is defined as F1= 1, F2= 1, Fn= Fn−1+ Fn−2(a) If you haven’t before, write out a few terms of this sequence to get a sense of the pattern.(b) Show that1Fn−1Fn+1=1Fn−1Fn−1FnFn+1(c) Show that∞Xn=21Fn−1Fn+1= 1(d) Show that∞Xn=2FnFn−1Fn+1=


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Berkeley MATH 1B - PDP Worksheet

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