80 Chapter 2 • Sequences
EXERCISE SET 2.3- Prove Corollary 2.3.2.
- In each of the following, find the limit and use the second squeeze principle
to prove that answer. (See Exercise 2.2.15.)
(a) lim -
6- = (b) lim -
1- =
n-+oo 2 + 7n n-+oo 4 - 5n
(c) · lim n =
n-+oo 3n + 8
(e) lim lOn -^11 =
n-+oo 7 - 2n
- =
(g) lim lOO =
n-+OO n2(i) lim _n_ =
n-+oo n^2 + 5
(k) lim 3n2 + n - 5
n-+oo n^2 + 6n(m) lim^5 n
n-+oo 2n3 - 7
n^3 - 2n^2
(o) lim
n-+oo 6 - 3n + 2n3- Prove Theorem 2.3.5 (b).
2n
(d) lim --=
n-+oo 3 - 4n( f) lim^5 -
2
n =
n-+oo 1+6n(h) lim
6
n =
n-+oo n4
(j) lim 8n + 9
n-+oo 8 - 3n^2
(1) lim 4n2 - 5n
n-+oo 8n^2 + 3n - 1
r 3n^2 + 7n - 4
(n) n2...1!, n3 - 42n^3 + 8n^2 - 23
(p) lim -----
n-+oo 5n3 - 6n- Prove Theorem 2 .3 .6 using Theorem 1.5.7 and the squeeze principle.
- Prove each of the following:
(a). 1 + 2 + 3 + · · · + n 1
hm = -
n-+oo n^2 2
. 12 + 22 + 32 + ... + n2
(b) hm 3
n--+oo n
1- 3
. 13 + 23 + 33 + ... + n3
(c) hm 4
n--+oo n
1- 4
[Hint: See Exercises 1.3.3- 1.3.5.]
- Geometric Series: (a) Given that lrl < 1, prove that
00
'"°'ark = lim (a + ar + ar^2 + · · · + arn) = a. [See the formula for
L n-+oo 1-r
k=O
finite geometric sums, Exercise 1.3.12.]
(b) Calculate lim (~+~+I_+···+ I_).
n-+oo 3 9 27 3n