Wednesday, 20 April 2016

real analysis - Show that sqrt2+sqrt2+sqrt2... converges to 2




Consider the sequence defined by
a1=2, a2=2+2, so that in general, an=2+an1 for n>1.
I know 2 is an upper bound of this sequence (I proved this by induction). Is there a way to show that this sequence converges to 2? What I think is that the key step is to prove 2 is the least upper bound of this sequence. But how?


Answer



Let x=2+2+2+. Then, note that x2=2+2+2+=2+xx2x2=0.Note that the two solutions to this equation are x=2 and x=1, but since this square root cannot be negative, it must be 2.


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