Exercise.
Prove that if
+1=
1+
where
0
and 10
, then the sequence
(
)
converges to the positive root of
2+=
.
NOTE: The word discuss
is used to frame deliberately open-ended
questions. Say as much as you reasonably can in the time available.
All your assertions should be proved. Discuss the convergence of
these sequences
clearly is asking for an indication of whether
the sequences do or do not converge, and if possible to what.
Solution to Exercise 2(a)
For
,
1+1
1+1+1
hence
(1+1
)
12
(1+1+1
)
12
so the sequence is monotonic decreasing. Each value
(1+1
)
12
is greater than zero
because 1+1
0 and the exponent 12
means take the positive square root, so the sequence is bounded
below by 0. Therefore by the Monotone Convergence Theorem,
for some
0
in .
By continuity of the square root function and 1+1
1
,
=(1+1
)
12=1
.
(Note: in this case the monotone convergence theorem was not
actually needed as the limit can be found and proved correct by
continuity. But in other cases such as 2(e) or 3(e) where the
limit is not so easy to evaluate you may like to use Monotone
Convergence.)