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

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    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.
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    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 View Post
    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 $\displaystyle l=\frac{1+\sqrt{1+4a}}{2}.$
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