Calculating the square root of x/sandbox

Step-by-step
Cut and paste the following wikitext into the sandbox, and enter your favourite number and a first (non-zero!) approximation. See what happen after multiple clicks of the "save" button: 17

Sandbox
100 10 10 10 10.000000000311 10.000078883626 10.039798887949 10.932861993477 15.351851851852 27 51.22 100 4.1231056256177 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310877528 4.1282051282 4.33333333333 3