Take 15 minutes to prepare this exercise.
Then, if you lack ideas to begin, look at the given clue and start searching for the solution.
A detailed solution is then proposed to you.
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Question
We denote by
a strictly positive real number and by
a real number belonging to interval
We define two real sequences
and
by:
and for any integer
greater or equal to
and
Find the limits of sequences
and
We can set
and look for the expressions of the sequences depending on
Express
and
in the form of a product of
To find the limit of
write it as quotient of
starting from
Let us assume:
The expressions of
and
are therefore justified by recursion for any natural integer
There remains to calculate the limit when
goes to infinity of
We obtain recursively:
We infer:
is equivalent to
when
goes to infinity, therefore:
goes towards
when
goes to infinity. This implies that both sequences converge towards
.