Can someone please help me with Lim excercise and explain it as well? Really appreciate!

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Answer 1 · 2018-04-18T22:38:10

#lim_(n->oo) u_n = 0#

Consider the sequence:

#u_1 = 1/4#

#(1) " " u_(n+1) = u_n^2+(u_n)/2#

If the sequence has a finite limit, then:

#lim_(n->oo) u_n = L#

but also:

#lim_(n->oo) u_(n+1) = L#

so if in the equation #(1)# we let #n->oo# we get:

#L = L^2+L/2#

#L^2=L/2#

which has two possible solutions:

#L=0#

and

#L=1/2#

From the definition we can now see that #u_1 > 0# and if #u_n > 0# then also #u_(n+1) > 0#. By induction we can therefore establish that #u_n > 0#.

Similarly we can see that:

#u_1 < 1/2#

and if #u_n < 1/2#, then:

#u_(n+1) = u_n^2+(u_n)/2 < 1/4+1/4 =1/2#

By induction then:

#u_n < 1/2# for #n >=1#

and then:

#u_(n+1) = u_n^2+(u_n)/2 < u_n/2+u_n/2 = u_n#

So the sequence is strictly decreasing and as #u_1 =1/4#, all its values are smaller than #1/4# and its limit cannot be #1/2#.

We can then conclude that:

#lim_(n->oo) u_n = 0#