Find the product of the first $20$ terms of the sequence below. Express your answer as a common fraction.

$\frac23, \frac34, \frac45, \frac56, \dots$
When the fractions are multiplied, many of the factors in the numerator and denominator will cancel, as it is a telescoping product:

$\frac23 \cdot \frac34 \dotsm \frac{21}{22}=\frac{2}{22}=\boxed{\frac{1}{11}}$.