A.
prove that for n 1
1 3
?
B.
prove that
and find the limit
?
it's a sequence not a serie.
Here are some hints:
A) Induction using the fact that the function defined by is increasing (because f'(x) is postive) then induction using
B) is increasing and bounded (is that the word because i don't study maths in english ) so it convergences then solve f(x)=x to compute the limit.
Hard luck .
We don't have to use Lagrange or Cauchy , those are simple sequences.
Let's stay with your problem:
A) we will use mathematical induction:
for n=1 we have 1=<2=<3 so 1=<a_{0}=<3
we assume that it's true for n and let's prove it for n+1
since 1 3 and f is increasing we have
End of induction .
B) let's prove that (a_{n}) is increasing :
we have now proved that it's increasing .
Now and since it's bounded , it converges.
As i said , to compute the limit , solve the equation f(x)=x
The 'initial value' is not specified and that isn't a minor detail. The 'recursive relation' can be written as...
(1)
The function f(*) is represented here...
There is only one 'attractive fixed point' at and that means that, if the sequence converges, it converges to 0. In particular the sequence converges monotonically for , converges 'with oscillation' for and diverges for and ...
Kind regards