# Let a>0 and let z1>0. Define zn+1:=for nN. show that (zn) converges and findthe limit

• Sep 19th 2012, 10:32 PM
TaTzRE66
Let a>0 and let z1>0. Define zn+1:=for nN. show that (zn) converges and findthe limit
Let a>0, and z1>0. Define zn+1= sqrt(a+zn), for n is a natural number. Show that {zn} converges and find the limit.
• Sep 19th 2012, 10:44 PM
FernandoRevilla
Re: Let a>0 and let z1>0. Define zn+1:=for nN. show that (zn) converges and findthe l
Quote:

Originally Posted by TaTzRE66
Let a>0, and z1>0. Define zn+1= sqrt(a+zn), for n is a natural number. Show that {zn} converges and find the limit.

1. Prove that $\displaystyle \{z_n\}$ is an increasing sequence.
2. Prove that $\displaystyle \{z_n\}$ has an upper bound.

According to a well known theorem, $\displaystyle \{z_n\}$ is convergent. If $\displaystyle l$ is its limit then, $\displaystyle l=\sqrt{a+l}.$ Solving this equation and taking into account that $\displaystyle z_n>0$ for all $\displaystyle n$, you'll find $\displaystyle l=\frac{1+\sqrt{1+4a}}{2}.$