an diverges. positive and decreasing sequence such that +ain- 39. Assume that (an) is a Show...
6. Suppose that {x,] is a sequence of positive numbers and limA = a Show that if L> 1 then lim x =00, and if L < 1 lim x = 0 n+02 b. Construct a sequence of positive numbers {x,} such that lim * = 1 and the sequence {x} diverges. c. Let k E N and a > 1 Show that lim = 0. O LIVE
Exercise 17: Let (an) be a sequence. a) Assume an> 0 for all n E N and lim nan =1+0. Show that an diverges. n=1 b) Assume an> 0 for all N EN and lim n'an=1+0. Show that an converges. nal
Determine the limit of the sequence or state that the sequence diverges. an = 3 3 n2 (Use symbolic notation and fractions where needed. Enter DNE if the sequence diverges.) lim an = n-00
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an > O for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
2. Let (%)-1 be a bounded sequence and let (h) .1 be a sequence that diverges to oo. Prove that (an +bn)ni diverges to oo
Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) a, = 5 + 8n2 " n + 8n2 lim n >00 an = Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) an = e-9/vñ lim n >00 an =
1. Let {an}, be a sequence. Write down the formal definition of the following con- cepts. You have already seen some of these in lecture (a) The sequence is convergent b) The sequence is divergent. (c) The sequence is divergent to oo (d) The sequence is divergent to -oo (e) The sequence is increasing f) The sequence is decreasing (g) The sequence is non-decreasing (h) The sequence isn't decreasing (i) The sequence is bounded above (j) The sequence is not...
3. Suppose lim s(a) dr = co, where f(a) is a positive, decreasing and continuous function. Which of the following statements is true about the series f(n)? Choose one. n=1 *Please write the letter of your choice. (a) The series converges too. (b) The series converges, but not necessarily to o. (c) The series diverges. (d) The given information is not enough to determine if the series converges or diverges.
1 7) Show that the series converges/diverges 1 1 8) Show that the sequence a N + 1 N+ 1 is monotonic.
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2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...