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## Homework Statement

find out for which values of ##\lambda>0## the sequence ##(a_n)## ,defined by

##a_1 = \frac{1}{2}, \quad\quad a_{n+1} = \frac{1}{2} (\lambda +a_n)^2, \quad n\in \mathbb{N^*}## converges.

If ##(a_n)## converges, find the limit.

## The Attempt at a Solution

If I assume ##a_n## converges, I can write the limit L instead of the elements of the sequence, such that:

## L=\frac{1}{2} \lambda^2 + \frac{1}{2} L^2 + \lambda L ##

and

## \lambda^2 + L^2 + 2\lambda L -2L=0##

Solving the equation, I have

## L_1=1-\lambda + \sqrt{1-2\lambda} ; L_2=1-\lambda - \sqrt{1-2\lambda} ##

##\lambda## must therefore be## ≤ \frac{1}{2}##

also, ##L_1## must equal ##L_2## due to the theorem of uniqueness of limits.

Thus, the only possible result is ##\lambda = \frac{1}{2}##.

And therefore ##L=\frac{1}{2}## too.