Sequence ?

Question

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!!!

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Answer 1 · 2017-02-05T16:42:50

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

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#