Question

Sequence ?

Hey guys!! sorry cz i didn't use LATEX.
Cz i really don't know how to use it.
Need help in the first exercise,
I have to find all values of a_1 such that the sequence is converge and i have to find the limit too.
Thanks alot.
https://files.acrobat.com/a/preview/b35761b1-baaf-4004-94b9-b140bef48ec4 the link of the exercise!!!

Answer 1

`AA a_1 in RR lim_(n->oo) a_n = a`

Explanation:

Given `a in RR`, the sequence `{a_n}` is defined as:

`a_(n+1) = a_n^2 +(1-2a)a_n +a^2`

Simplify:

`a_(n+1) = a_n^2 +a_n-2a*a_n +a^2 = a_n+ (a_n -a)^2`

Passing to the limit:

`lim_(n->oo) a_(n+1) = lim_(n->oo) a_n + lim_(n->oo) (a_n-a)^2`

But if the series is convergent:

`lim_(n->oo) a_(n+1) = lim_(n->oo) a_n = L`

so:

`lim_(n->oo) (a_n - a)^2 = 0`

and also:

`sqrt(lim_(n->oo) (a_n - a)^2) = 0`

but since `sqrt(x)` is a continuous function:

`sqrt(lim_(n->oo) (a_n - a)^2) = lim_(n->oo) sqrt ((a_n - a)^2) =lim_(n->oo) abs(a_n-a) = 0`

That is:

`lim_(n->oo) a_n = a`

independently of how we choose `a_1`