To prove that the sum of 1/20 + 1/21 + 1/22 +...+ 1/59 is less than 3/2, we can first find a common denominator for all the fractions in the series.

The common denominator for all the fractions from 1/20 to 1/59 is 20*21*22*...*59. We can rewrite each fraction with this common denominator:

1/20 = (20*21*22*...*59) / (20*21*22*...*59*20)
1/21 = (20*21*22*...*59) / (20*21*22*...*59*21)
...
1/59 = (20*21*22*...*59) / (20*21*22*...*59*59)

Now, we can add all these fractions together:

(20*21*22*...*59) / (20*21*22*...*59*20) + (20*21*22*...*59) / (20*21*22*...*59*21) + ... + (20*21*22*...*59) / (20*21*22*...*59*59)

= (20*21*22*...*59) * (1/20 + 1/21 + 1/22 + ... + 1/59)

Since the sum of the fractions is less than 3/2, we have:

(20*21*22*...*59) * (1/20 + 1/21 + 1/22 + ... + 1/59) < (20*21*22*...*59) * (3/2)

Therefore, 1/20 + 1/21 + 1/22 +...+ 1/59 < 3/2.