Adding Series of Such Fractions as a/b+b/c+c/d+d/e+...

1/2+2/3+3/4+4/5+...+n/n+1 = ?
(n/n+1)+(n+1/n+2)+(n+2/n+3)+... = ?
(0.00...01/0.00...01+1)+(0.00...01+1/0.00...01+2)+... = ?
(0.00...01/1)+(1/2)+(3/4)+(4/5)+... = ?

a/b+b/c = (ac+b²)/bc
a/b+b/c+c/d+d/e = ((ac+b²)/bc)+((ce+d²)/de) = (de(ac+b²)+bc(ce+d²))/bcde = (acde+bc²e+b²+d²)/bcde
a/b+b/c+c/d+d/e+e/f+f/g+g/h = 

try if you like. I haven't found the logic behind it yet. Have no time for it. If you've got it, share it, please.