To find the sum of the series:

1/1 5 + 1/5 9 + ... + 1/2n(2n+2)

First, let's simplify each term:

1/1 5 = 5
1/5 9 = 9/5
...

To find a general formula for the nth term, we notice that the pattern for each term is (1/n) * (2n+1).

Therefore, the nth term of the series is:
1/n * (2n+1)

Now, we can express the sum of the series as follows:

Sum = (1/1 (21+1)) + (1/2 (22+1)) + ... + (1/n * (2n+1))

Sum = Σ(1/n * (2n+1)) for n = 1 to n

Expanding the summation gives:

Sum = (1 3/1) + (2/2 5) + ... + (n/n * (2n+1))

Sum = 3 + 5 + ... + (2n + 1)

Now, we can simplify this:

3 + 5 + ... + (2n + 1) = (2n + 1)(2n + 3)/2

Therefore, the sum of the series 1/1 5 + 1/5 9 + ... + 1/2n(2n+2) is:

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