Question #90ab5

1 Answer
NickTheTurtle
Jan 23, 2018

#lim_(n->oo)a_n=2#

Explanation:

#a_(n+1)=sqrt(2+a_n)#

Assuming that #a_n# exists when #n->oo#, then
#lim_(n->oo)a_(n+1)=lim_(n->oo)sqrt(2+a_n)#
#lim_(n->oo)a_n=lim_(n->oo)sqrt(2+a_n)#
#lim_(n->oo)a_n^2=lim_(n->oo)2+a_n#
#lim_(n->oo)a_n^2-a_n-2=0#
#lim_(n->oo)a_n=(1±sqrt(1^2+4*2))/2=(1±3)/2#

However, #a_n>0# for all #n#. Thus,
#lim_(n->oo)a_n=2#

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