Can you explain how this code (recursive formula) works?

Question

Mathematics

Number 4.
Here is the code:

program Project1;

{$APPTYPE CONSOLE}

uses
SysUtils;

var
x, eps, R, S, St, Ea, Eo: real;
n, k: integer;

begin
writeln('Ведите x, eps, k');
readln(x, eps, k);
writeln;
n:=0;
S:=1+(x*x)/2;
R:=1*(x*x)/2*2;
St:=sqrt(1+x);
while abs(R)>=eps do
begin
n:=n+1;
R:=(-1)*R*x*(2*n-1)/(2*n+2);
S:=S+R;
end;
Ea:=(Abs(sqrt(1+x)-S));
Eo:=(Abs((sqrt(1+x)-S)/sqrt(1+x)));
writeln('S=', S:k+3:k);
writeln('R=', R:k+3:k);
writeln('n=', n);
writeln('St=', St:k+3:k);
writeln('Ea=', Ea:k+3:k);
writeln('Eo=', Eo:k+3:k,'%');
readln;
{ TODO -oUser -cConsole Main : Insert code here }
end.

I don't understand the math here.
I can make a similar program, but using conditional statements. Here it all works with one formula.