Write the sum using sigma notation: \[ \begin{array}{l} \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{209 \cdot 210}=\sum_{n=1}^{A} B, \text { where } \\ A=\square \\ B=\square \end{array} \]

Step 1 ：The given series is a sequence where each term is the reciprocal of the product of two consecutive integers. The pattern suggests that the series starts from n=1 and ends at n=209.

Step 2 ：Therefore, A should be 209.

Step 3 ：The general term B of the series can be written as 1/(n*(n+1)).

Step 4 ：Let's write this in sigma notation.

Step 5 ：Final Answer: \(\boxed{A=209}\), \(\boxed{B=\frac{1}{n \cdot (n+1)}}\)

Step 6 ：So, the sum in sigma notation is \(\sum_{n=1}^{209} \frac{1}{n \cdot (n+1)}\)