Question

Show that the sequence an defined by: a1= √ 2 & an=√ 2+an-1,n>1 converges

Answer 1

\(a_n = \sqrt{2+a_{n-1}}\), right?

In that case, \(a_2 = \sqrt{2+\sqrt2},~a_3 = \sqrt{2+\sqrt{2+\sqrt2}},~\cdots\)

So \(S = \displaystyle\lim_{n\to\infty}a_n \Longrightarrow S = \sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=\sqrt{2+S}\)

So you have \(S = \sqrt{2+S}\). Solve for S, then you can see that as n approaches to infinity, S approaches to certain value, hence it converges.

Hope that helps.

Answer 2

Its like inception