Question

the sequence {a_n} is defined by a_1= 1 and a_(n+1) = 1/2( a_n+ (4/a_n) ) for >=1. Assuming that {a_n} converges, find its limit.

Answer 1

`a_(n+1)=\frac{a_n+\frac4{a_n}}2`

Assuming convergence of the sequence, we have
`a_n\to L` for some limit
`L` as
`n\to\infty`. This also means
`a_(n+1)\to L`. So in the above equation, we can substitute and solve for
`L`:

`L=\frac{L+\frac4L}2`

`\implies2L^2=L^2+4`

`\implies L^2=4\implies L=\pm2`

The limit can only be one of these values; otherwise, the sequence would be divergent. To decide which one is correct, we can observe that the sequence is strictly positive. So
`L=2`.