Download Math 1B PDP Worksheet: Sequences and Limits and more Assignments Calculus in PDF only on Docsity! Rob Bayer Math 1B PDP Worksheet March 3, 2009 Sequences 1. For each of the following sequences, write the form of the general term an, starting your indexing at n = 1. Also determine whether each sequence is convergent or divergent. For those that are convergent, find the limit. (a) {1, 2, 3, 4, . . .} (b) {2,−2, 2,−2, 2, . . .} (c) {4, 7, 10, 13, . . .} (d) { 12 ,− 1 4 , 1 8 ,− 1 16 , . . .} (e) {− 12 , 2 3 ,− 3 4 , 4 5 , . . .} 2. Determine whether each of the following sequences are convergent or divergent. For those that are convergent, find the limit. (a) an = 3n 2+1 n2−1 (b) an = (n+2)! n2·n! (c) an = cos 2πn (d) {1, 12 , 1, 1 4 , 1, 1 8 , . . .} (e) an = ln(n2 − 3n+ 1)− ln(n2 + 4) (f) an = sin nn (g) an = n tan(1/n) 3. Consider the sequence an = rn, where r is a constant. (a) Write out a few terms of this sequence for r = −2,−1, 1/2, 1, 2. What is the limit in each of these cases? (b) In general, for what values r does this sequence converge? Find the limit for those values. (c) Repeat part (b) for the sequence an = nrn. 4. Let p(x) = blxl + bl−1xl−1 + · · ·+ b0, q(x) = cmxm + cm−1xm−1 + · · ·+ c0 be polynomials of degrees l,m respectively. Define a sequence an by an = p(n) q(n) . Determine whether limn→∞ an exists in each of the following cases. When it does, find its value. (a) deg p < deg q (b) deg p = deg q (c) deg p > deg q 5. True/False. For all problems, an and bn are sequences. Justify your answers with a sketch of a proof or a counterexample. (a) If an and bn converge, then an + bn converges. (b) If an + bn converges, then an and bn converge. (c) If an and bn converge, then an/bn converges. (d) If an and bn diverge, then an + bn diverges. (e) If an + bn diverges, then an and bn diverge. (f) If an and bn diverge, then anbn diverges. The −N definition of a limit 1. (a) Prove, using the −N definition, that if lim n→∞ |an| = 0, then lim n→∞ an = 0. (b) Find a counterexample to the statement “If lim n→∞ |an| = L, then lim n→∞ an = L” 2. Prove, using the −N definition, that lim n→∞ sinn n = 0 3. (hard) Prove, using the −N definition, that if lim n→∞ an = L, lim n→∞ bn = K then lim n→∞ (an + bn) = L+K
