Calculus II – Worksheet 11
8.1 Sequences
1. Find an explicit formula for the sequence {1,−1, 1,− 1 , 1 ,…}, assuming that n begins with 1.
39 2781
2. Determine the limit of the sequence or state the limit does not exist. If there is a limit, show the calculations that explain how you are finding the limit. Determine whether the sequence is increasing, decreasing, not monotonic, and/or bounded.
n4
(a) n4+1 forn≥1
cosn
(b) √
for n ≥ 1
3. Is the sequence a = n! for n ≥ 1 convergent?
4. Determine for what values of r is {rn}∞n=1 convergent, and what does it converge to. For what values of r is it divergent?
n
n nn
Extra Practice
5. Find an for each of the sequences below
(a) (b)
(c)
(d) (e) (f)
(g)
{1,4,9,16,…} 1111
2, 5, 8, 11,… 1111
2, 4, 8, 16,… −35−79
2 ,4, 8 ,16,… {0,2,0,2,…}
4 6 8 10 1, 7, 12, 17, 22,…
5 8 11 14 3, 9, 27, 81,…
6. Determine whether the sequences in problem 1 converge or diverge, and find the limits for the ones that do converge.
7. Prove that the sequence an = n , n = 1,2,3,… is decreasing. n2 + 1
8. Determine the limit of the sequence or state the limit does not exist. If there is a limit, show the calculations that explain how you are finding the limit.
(a) e7/n for n ≥ 1 6n
(b) 1 + n for n ≥ 1