Prove that the following sequence is convergent: `u_1=sqrt(2),` `u_2=sqrt(2+sqrt(2)), ldots, u_n=sqrt(2+sqrt(2+sqrt(2+sqrt(2+cdots+sqrt(2+sqrt(2))))))`
In order to prove that your sequence is convergent we will use the following theorem:
A sequence of real numbers is convergent if it is monotone and bounded.
1. proof of monotonicity
We will prove this by induction:
i) `x_2=sqrt(2+sqrt(2))>sqrt(2)=x_1` which proves the base.
ii) Assume that for every `k leq n` `x_k>x_(k-1).`
iii) Now we use assumption for `k=n` and add number 2 to each side.
`x_(n+1)=sqrt(2+x_n)>sqrt(2+x_(n-1))=x_n` which proves inductive step.
2. proof that sequence is bounded
Since sequence is increasing we only need to find upper bound (lower bound is `x_1` ). We will prove that the upper bound is 2. Again we do this by induction:
ii)Assume that `x_n<2.`
iii) `x_(n+1)=sqrt(2+x_n)<sqrt(2+2)=2` which proves step of induction.
We have now proven that the sequence is convergent.
We can also find the limit of the sequence:
We know that sequence is convergent meaning it has limit `a.` This means:
`lim_(n rightarrow infty) x_n=lim_(n rightarrow infty) sqrt(2+x_(n-1))`
by squaring this equation we get
`a^2-a-2=0` solutions to this equation are `a_1=2` and `a_2=-1.` Since only `a_1=2` satisfies (1) that is our limit.