Math304 Test Quiz 1–Version B sec-SA,SB Instructions. (24 points) Solve each of the following problems.
(10pts) 1.
(a) (5 pts) Determine whether the sequence an = en+ 1
n+ 2n converges or diverges. Justify your answer.
Solution:
n→∞lim an= lim
n→∞
en+ 1
n+ 2n dived up and down by 2n.
= lim
n→∞
en 2n + 1
2n n 2n +2n
2n
simplify.
= lim
n→∞
(e2)n+ (12)n
2nn + 1 (e/2)
n→ ∞,(1/2)n, n/2n→0 asn→ ∞.
=∞ Hence lim
n→∞an=∞and hence the sequence diverges.
(b) (5 pts) Determine whether the sequence an =n+ 1
n+ 2 is increasing, decreasing, or neither.Justify your answer.
Solution:
Letf(x) = x+ 1
x+ 2, x≥1.
f(x) = (x+ 1)(x+ 2)−1, x≥1 f(x) = (x+ 2)−1−(x+ 1)(x+ 2)−2
= (x+ 2)−2[x+ 2−x−1]
= 1
(x+ 2)2
≥0 for allx≥1
Hence the sequence is increasing.
Math304 Test Quiz 1–Version B sec-SA,SB (4pts) 2. Put (T) if the statement is true and (F) if the statement is false.Justify your answer.
(a) (1 pts)F Every convergent sequence is monotonic .
Solution: The statement is false since the sequence an = (−1)n
n is convergent but not monotonic.
(b) (1 pts)T The series
∞
n=1
nis divergent .
Solution: The statement is true. Since lim
n→∞n = ∞ = 0, then
∞
n=1
n is divergent by Divergence Test .
(c) (1 pts)F The sequence
n+ 3
n
n
converges to e−3. Solution: The statement is false. Since
n→∞lim n+ 3
n n
= lim
n→∞
1 + 3
n n
=e3.
(d) (1 pts)T The sequence{tan−1n} is increasing.
Solution: The statement is true. Now, let f(x) = tan−1x, ∀x ≥ 1. Then f(x) = 1
1 +x2 >0, ∀x≥1.Hence The sequence {tan−1n}is increasing.
Math304 Test Quiz 1–Version B sec-SA,SB (10pts) 3.
(a) (5 pts)Test the given series for convergence or divergence. State the test you used to get your answer.
∞
n=1
nsin
1
n
. Solution:
Letan=nsin
1
n
.Now,
n→∞lim an = lim
n→∞nsin
1
n
= lim
n→∞
sin1
n
n1
= 1= 0.
Hence the series is divergent by Divergence Test
(b) (5 pts) Give an example for: An increasing sequence that is divergent.
Solution:
The sequencean=nis an increasing sequence that is divergent.
