Take 15 minutes to prepare this exercise.
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A detailed solution is then proposed to you.
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Question
For any natural integer
greater or equal to
we define the function
from
to
by:
a) Prove that equation
has a unique solution in
We denote this unique solution by
b) Study the variations of sequence
c) Find the limit of sequence
when
goes to infinity.
d) Find an equivalent of sequence
when
goes to infinity.
a) Study the variations of application
on
and apply the bijection's theorem.
b) Find the sign of
then use the array of variations of
to place
with respect to
c) Prove, using the monotony of the sequence, that
converges towards
d) Use the limit of
determined at the previous question.
a) We define a function
by
Function
is differentiable and
The derivative of function
is therefore strictly decreasing on
Moreover
and
which is strictly negative for
greater than
hence by the bijection theorem, there exists a unique element
of
such that
b) By the table of variations of function
if
is positive,
and if
Hence we study the sign of
and
by definition of
We infer from this last equality:
which gives us:
which is positive since
We infer:
and the sequence is decreasing.
c) Sequence
is decreasing and has a lower bound (by
) hence its is convergent.
We have, for any natural integer
greater or equal to
We also have
which implies that
and
goes towards
when
goes to infinity, hence by the dual inequality,
goes towards
when
goes to infinity.
We infer that
goes towards
when
goes to infinity, which implies that
goes towards
when
goes to infinity.
d) We saw in the previous question that
goes towards
when
goes to infinity, which gives directly
when
goes to infinity.
