MA1S12: SOLUTIONS TO TUTORIAL 7
1. Write the nth term un of the following sequences and state if they
are convergent, divergent, bounded or monotone.
−1, 1, −1, 1, −1 . . .
1, 3, 5, 7, 9 . . .
Solution: The first sequence has nth term un = (−1)n . This is a
bounded sequence since |un | ≤ 1 for all n. The sequence is divergent
and not monotone.
The second sequence has nth term un = 2n−1. This is a monotone
(increasing) sequence. The sequence is unbounded and divergent.
2. Write the first three terms of the following sequences {un }∞
n=1 and
determine if they are convergent or divergent.
un =
n+2
n
un =
n+1
2n2 + 2
The sequence H-1L^n is divergent
1.0
0.5
5
10
-0.5
-1.0
1
15
20
2
MA1S12: SOLUTIONS TO TUTORIAL 7
The sequence 2n-1 is divergent
30
20
10
5
10
15
20
Solution: Both of these sequences are convergent. The first three
terms of the first sequence are
u1 = 3,
u2 = 2,
u3 =
5
3
To find the limit we divide the numerator and denominator by the
highest power of n,
1+
n+2
= lim
n→∞
n→∞
n
1
lim
2
n
=1
The first three terms of the second sequence are
1
u1 = ,
2
u2 =
3
,
10
u3 =
1
5
Again to find the limit we divide the numerator and denominator
by the highest power of n,
lim
n→∞
1
+ n12
n+1
n
=
lim
=0
2n2 + 2 n→∞ 2 + n22
3. Find the limit of the following sequences
1
3n
∞
n=1
(−1)n
n
∞
n=1
MA1S12: SOLUTIONS TO TUTORIAL 7
3
The sequence Hn+2Ln converges to 1
2.0
1.5
1.0
0.5
5
10
15
20
The sequence Hn+1LH2n^2+2L converges to 0
0.30
0.25
0.20
0.15
0.10
0.05
5
10
15
20
Solution: Both of these sequences are convergent. For the first sequence note that for all n we have
0≤
1
1
≤
3n
n
Also
1
=0
n→∞ n
lim 0 = lim
n→∞
and so by the Squeezing theorem,
1
=0
n→∞ 3n
lim
For the second sequence note that
−
1
(−1)n
1
≤
≤
n
n
n
4
MA1S12: SOLUTIONS TO TUTORIAL 7
13^n is squeezed between 0 and 1n
1.0
à
0.8
0.6
à
0.4
æ
à
à
à
0.2
à
æ
æ
2
æ
4
æ
æ
6
à
à
à
à
æ
æ
8
æ
æ
10
H-1L^nn is squeezed between -1n and 1n
0.4
0.2
5
10
15
20
-0.2
-0.4
We have
lim −
n→∞
1
1
= lim = 0
n n→∞ n
and so by the Squeezing theorem,
(−1)n
lim
=0
n→∞
n
4. Determine if the following sequences converge or diverge
n 1 o∞
5n
n=1
n+1
en
∞
n=1
Solution: Both of these sequences are convergent. For the first sequence recall that if f (x) is a continuous function and un → L then
f (un ) → f (L). In this question we can use the continuous function
MA1S12: SOLUTIONS TO TUTORIAL 7
5
The sequence 5^H1nL converges to 1
2.5
2.0
1.5
1.0
0.5
5
f (x) = 5x . We know
1
n
10
15
→ 0 and so
1
lim 5 = lim f
= f (0) = 50 = 1
n→∞
n→∞
n
1
n
For the second sequence we use the fact that if a function f (x)
converges to L as x → ∞ and if a sequence {un }∞
n=1 has nth term
un = f (n) then the sequence {un }∞
n=1 also converges to L. In this
question we take f (x) =
x+1
.
ex
Note that
lim (x + 1) = ∞
x→∞
lim ex = ∞
x→∞
and so by L’Hôpital’s rule
x+1
1
= lim x = 0
x
x→∞ e
x→∞ e
lim f (x) = lim
x→∞
Thus the sequence also converges,
n+1
= lim f (n) = 0
n→∞ en
n→∞
lim
Alternatively, we could show that both sequences are monotone
(decreasing) and bounded and so by the Monotone Convergence
Theorem they are convergent.
6
MA1S12: SOLUTIONS TO TUTORIAL 7
The sequence Hn+1Le^n converges to 0
æ
0.7
0.6
0.5
0.4
æ
0.3
0.2
æ
0.1
æ
æ
2
4
æ
6
æ
æ
8
æ
æ
10
æ
æ
12