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and hence
2
3 a
1
a
2
a .
(c) Deduce that, eventually,
1
a
1
a
2
a ยท a a
.
a
(d) Now use (a), the scalar rule for null sequences (qn 32) and the
squeeze rule with a shift (qn 46(ii)) to prove that (1/ a
)
1/ a .
66 Let ( a
) be a sequence of non-zero terms with ( a
) a and a 0.
Provethat (1/ a
) 1/ a , by applying thersult of qn 65 to the
sequence (
a
).
67
The quotient rule
Let ( a
) a ,( b
) b , and supposethat all of the b
and b are
a / b .
non-zero. Prove that ( a
/ b
)
68 Find the limits of the sequences with n th terms given here, stating
which rules you are using.
(i) 1 1/ n
(ii) 3 n 2
(iii) n 1
(iv) (
) 1
2 1/ n ,
4 n 3 ,
3 n n ,
) 1 ,
(
(v) 2 1
(vi) n n n 1
3 n 5
(vii) ( n 3)( n 1)
3 n 5 n
2 1 ,
,
,
(viii) a 1
a 1 ,0 a , (ix) 1 2 ... n
(x) b , where 0 b 1.
,
n
69 Let a
( n 8) n , b
( n n ) n and
c
( n n /8)
n .
Show that a
when n 64.
Find the limits of the sequences ( a
b
c
), ( b
) and ( c
) if they exist.
( Hint . For positivenumbrs, x and y ,
x y ( x y )( x y ).
d'Alembert's ratio test
70 Give examples of sequences ( a
) of positivetrms for which
a
a
.
You will find one example in qn 40. Are all the sequences which
you have found convergent? To what limit?
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