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Berkeley MATH 1B - PDP Worksheet

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Rob Bayer Math 1B PDP Worksheet March 10, 2009More Sequences1. Determine whether each of the following sequences are monotone, bounded, both, or neither.(a)13n − 4(b) sin(1n)(c)3n − 2n + 12. Give an example of each of the following:(a) A bounded sequence that diverges(b) A monotone sequence that diverges(c) A convergent sequence that isn’t monotone3. In addition to defining sequences by a formula of the form an= f(n), we can also define a sequence by givingthe value of a1and a rule for getting the next term in the sequence from the previous one. For example, wecould say an= 3 + an−1; a1= 2. This then defines the sequencea1= 2, a2= 3 + a1= 3 + 2 = 5, a3= 3 + a2= 8, a4= 3 + a3= 11, etc. Such a definition is called recursive. Adefinition that doesn’t refer to previous terms or have any “...”s in it is called a closed form. For example, aclosed form for the sequence above would be an= 2 + 3(n − 1)For each of the following recursively defined sequences, find a closed form for an. Use your answer to determineif each is convergent or divergent.(a) an= nan−1; a1= 1(b) an= an−1+ n; a1= 1(c) an= ran−1; a1= a where a, r are contants and |r| < 1(d) an= ln n + an−1; a1= 0Series I1. Re-write each of the following inPnotation. Start your sum wherever is convenient.(a) 1 + 2 + 3 + ···(b) 1 +14+19+ ···(c) 1 − 1 + 1 − 1 + ···(d)32−34+38−316+ ···2. Re-index each of the following series to start at n = 0(a)∞Xn=1arn−1(b)∞Xn=2lnn + 1n + 2(c)∞Xn=−1sinnnπ23. Determine whether each of the following series are convergent or divergent. For those that are convergent, findthe sum.(a)∞Xn=13n+222n(b)∞Xn=1n√2 (c)∞Xn=0(−1)n2 + 3n4n(d)∞Xn=1n2− 3n2+ 3n + 14. By using partial sums (ie, the definition of a series), determine whether each of the following converge ordiverge. For those that converge, find the sum:(a)∞Xn=1sin(n + 1) − sin(n) (b)∞Xn=2lnn2+ 2n + 1n2(c)∞Xn=1−2n(n + 2)5. If the nth partial sum of a series∞Xn=1anis sn=n − 1n + 1, find anand∞Xn=1an6. For which values of x do each of the following converge?(a)∞Xn=0xn(b)∞Xn=0xn2n(c)∞Xn=1xn(d)∞Xn=0(x + 1)n32n+17. True/false. For those that are true, provide a brief explanation/intuition of why. For those that are false, find acounterexample:(a) If anis positive for all n, and each partial sum is less than 104, thenP∞n=0anconverges(b) If an< bnfor all n and both sequences converge, then lim an< lim bn(c) If snis the sequence of partial sums for the sequence anand limn→∞an= 0, then


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Berkeley MATH 1B - PDP Worksheet

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