/sci/ - Science & Math

>>14871182
Note:
[math] x = \tfrac{1}{2}x + \tfrac{3}{2}/x \iff 2x^2 = x^2 + 3 \iff x = \sqrt{3} [/math]
So if it converges at all, the fixed point of the series should be sqrt(3).

From another angle:
[math] a_{n+1} = \tfrac{1}{2}a_n + \tfrac{3}{2}/a_n [/math]
[math] a_{n+1} = a_n \left( 1 +( \sqrt{3}/a_n)^2 \right) / 2 [/math]

So also here the fixed point is clear.

Leaves to find some argument for why the starting point 1 converges. I'm sure there's both standard sequence arguments as well as a more tailor made fixed pont convexity case to be made for it