I. Show that the series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ is convergent and find its sum. [20 points]

Step 1 ：The series given is a telescoping series. A telescoping series is a series where each term u_k can be written as u_k = t_k - t_{k+1} for some sequence {t_k}. The sum of the first N terms of a telescoping series is then S_N = t_1 - t_{N+1}. If the sequence {t_k} converges to some limit L as k goes to infinity, then the series is convergent and its sum is S = t_1 - L.

Step 2 ：In this case, we can write \(\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}\). So the series is a telescoping series with t_n = 1/n.

Step 3 ：The sequence {1/n} converges to 0 as n goes to infinity, so the series is convergent and its sum is S = 1 - 0 = 1.

Step 4 ：Final Answer: The series \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\) is convergent and its sum is \(\boxed{1}\).