HW 2
JIA-MING (FRANK) LIOU
1. Part I
(1) Determine whether the series converges or diverges. Write c if it is convergent and d if it is divergent. Explain your answer.
(a)
∞
X
n=1
sin π n2. (b)
∞
X
n=1
n!
nn. (c)
∞
X
n=1
(−1)n−1 1 2n + 1. (d)
∞
X
n=1
cos nπ 2n . (e)
∞
X
n=1
n + 2 (n + 1)3. (f)
∞
X
n=1
(2n − 1)(n2− 1) (n + 1)(n2+ 4)2.
(2) Determine whether the series is absolutely convergent, conditionally convergent or divergent. Explain your answer.
(a)
∞
X
n=1
(−1)n−1
√4
n . (b)
∞
X
n=1
n 2 3
n
.
(c)
∞
X
n=1
1 + 1
n
n2
.
(d)
∞
X
n=1
(−3)n (2n + 1)!. (e) X
n=1∞
(2n)!
n!n!. (f)
∞
X
n=1
n2+ 1 2n2+ 1
n
1
2 JIA-MING (FRANK) LIOU
(3) Let hn = √n
n − 1 for n ≥ 1. In this exercise, we are going to study whether the alternating series
∞
X
n=1
(−1)n−1hnis absolutely convergent or conditionally convergent.
(a) Show that
0 < hn<
r2
n, n ≥ 2
(Hint: consider (1 + hn)n.) By the Sandwich principle, we find lim
n→∞
√n
n = 1.
(b) Show that for n ≥ 3,
nn+1> (n + 1)n. Using this, show that (hn) is decreasing.
(c) From (a) and (b), we know that the series
∞
X
n=1
(−1)n−1hn is convergent by the Leibnitz test. Is the series absolutely convergent?
