Download W.14 Limit of Sequence

W.14 Limit of Sequence
1. Let M be a real number greater than 1 . A sequence
a n+1 =
M2 + a n
1 + an
 an  is defined by
a1 > M and
for n  1 .
(a) By expressing a2n+1 in terms of a2n-1 , show that the sequence { a1 , a3 , a5 , … } is monotonic
decreasing.
(b) Show that the sequence { a1 , a3 , a5 , … } is convergent and find its limit .
2.
Let a be a positive real number .
(a) Show that the equation a x2 + 2x - a = 0 has two real roots of opposite signs .
(b) Let a1 = a and for n > 1, an+1 be the positive real roots of the equation an x2 + 2x - an = 0 .
(i) By expressing an+1 in terms of an , show that the sequence { an } is monotonic decreasing.
(ii) Hence, show that the sequence { an } is convergent and find its limit .
3.
Let { an } be an increasing convergent sequence of positive numbers with the limit value equal to a .
1
Suppose { bn } and { cn } are two sequences defined by b1 = c1 =
a and
2 1
bn =

1
a
+ cn-1
2 n-1

,
cn =
a n-1 bn-1
for n  1 .
(a) Show by induction that
(i)
(ii)
both { bn } and { cn } are strictly increasing sequences ,
for n  1 .
bn < an and cn < an
(b) Show that both the sequences { bn } and { cn } are convergent .
Hence , evaluate
4.
lim bn
n
and
1
=
(a) By using the result 1 n
lim cn .
n
1
1
1+
n-1
1

, show that
lim  1 - 
n
n 
n
= e .
(b) By using the result in (a) , evaluate
n
(i)
1

lim 1 
3n 
n 
(iii)

2
8
lim  1 + - 2 
n n 
n 
2 n
(ii)
4 

lim  1 
2n - 3
n 
(iv)

3
8 

lim  1 4n 8 n 2 
n 
n
n
.
n
5.
Let xn =

k=0
1
k!
1

and yn = 1 + 

n
n
for n  1 .
(a) Show that for any positive integer n ,
(i)
xn < xn+1
(ii)
xn < 3 .
(b) Using binomial theorem , show that
n
1 
1 
2
r - 1

yn = 2 +
(i)
 1 -   1 -  ...  1 

r!
n 
n
n 
r=2

(ii)
yn  xn
for n  2 ,
for all positive integers n .
(c) If n is fixed and greater than 1 , show by induction that for each positive integer r ,
1 -
r (r - 1)
n

1 
2
r - 1


1 -  1 -  ... 1 


n 
n
n 
Hence , show that for n > 1 ,
(d) Show that
lim xn
n
=
1

 1 -  xn  yn .

n
lim yn .
n
where 2  r  n .