The problem is:
Prove the convergence of the sequence
$\sqrt7,\; \sqrt{7-\sqrt7}, \; \sqrt{7-\sqrt{7+\sqrt7}},\; \sqrt{7-\sqrt{7+\sqrt{7-\sqrt7}}}$, ....
AND evaluate its limit.
If the convergen is proved, I can evaluate the limit by the recurrence relation
$a_{n+2} = \sqrt{7-\sqrt{7+a_n}}$.
A quickly find solution to this quartic equation is 2; and other roots (if I find them all) can be disposed (since they are either too large or negative).
But this method presupposes that I can find all roots of a quartic equation.
Can I have other method that bypasses this?
For example can I find another recurrence relation such that I dont have to solve a quartic (or cubic) equation? or at least a quintic equation that involvs only quadratic terms (thus can be reduced to quadratic equation)?
If these attempts are futile, I shall happily take my above mathod as an answer.
