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.
If you have more questions, feel to ask then on the forum.
Question
We denote by
a non zero natural integer and we define an application
from
to
by:
Find
Set
then make a Taylor-expansion of order
of
and get back to a form
Application
has strictly positive values, hence we can set
For any natural integer
Since
goes towards
when
goes to
we can use a Taylor-expansion of the exponential application, and we obtain:
We infer:
Since
goes towards
when
goes to
we obtain by using a Taylor-expansion of
therefore:
and the exponential application is continuous, therefore:
