Build general form of an infinite sequence

FindSequenceFunction[{8/35, 5/21, 8/33, 35/143, 16/65, 21/85, 
                      80/323, 33/133, 40/161}, n] // FullSimplify

(*    ((1 + n) (3 + n))/((3 + 2 n) (5 + 2 n))    *)

Using SequenceToSum:

 ResourceFunction["SequenceToSum"] [{8/35, 5/21, 8/33, 35/143, 16/65, 21/85, 80/323, 33/133,40/161, \[Ellipsis]}, n]

 (*Inactive[Sum][(3 + 4 n + n^2)/(15 + 16 n + 4 n^2), {n, 1, \[Infinity]}]*)

$$\underset{n=1}{\overset{\infty }{\sum }}\frac{n^2+4 n+3}{4 n^2+16 n+15}$$