
Infinite Sequences and Series 17-11
Example 17.12 Show that the sequence a b
n n
n
+
{ }
1 /
is convergent where a, b Î N such that a > b.
Also find the limit point of the sequence.
Solution: Since b > a, we have
b
n
< a
n
+ b
n
and a
n
+ b
n
< a
n
+a
n
= 2a
n
.
Thus,
b a b a
b a b a
b a b
n n n n
n n n n n n n
n n n n
< + <
⇒ < + <
⇒ < + <
2
2
2
1 1 1
1 1
( ) ( ) ( )
( )
/ / /
/ /
aa a< 2 .
Therefore, the sequence
a b
n n
n
+
{ }
1 /
is bounded.
Also if we let
s a b
n
n n n
= +( )
/1
then s a b
n
n n n
+
+ + +
= +
1
1 1 1 1
( ) .
Now,
( ) ( )
/ /
a b a a a a a a
n n n n n n n n n
+ > = =
+1 1 1
(1)
Also
( ) ( )
/ /
a b b b b bb b
n n n n n n n n n
+ > = =
+1 1 1
(2)
Adding Eqs. (1) and (2), we obtain ...