To find the sum of the series \( \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \frac{1}{42} \), we need to find a common denominator and then add the fractions.

First, let's find the least common multiple (LCM) of the denominators: 6, 12, 20, 30, and 42.

- Prime factorization of 6: \( 2 \times 3 \)
- Prime factorization of 12: \( 2^2 \times 3 \)
- Prime factorization of 20: \( 2^2 \times 5 \)
- Prime factorization of 30: \( 2 \times 3 \times 5 \)
- Prime factorization of 42: \( 2 \times 3 \times 7 \)

The LCM is found by taking the highest power of each prime that appears in the factorizations:
- \( 2^2 \) (from 12 and 20)
- \( 3 \) (from 6, 12, 30, and 42)
- \( 5 \) (from 20 and 30)
- \( 7 \) (from 42)

So, the LCM is \( 2^2 \times 3 \times 5 \times 7 = 4 \times 3 \times 5 \times 7 = 420 \).

Now, we convert each fraction to have the denominator 420:

\[
\frac{1}{6} = \frac{70}{420}
\]

\[
\frac{1}{12} = \frac{35}{420}
\]

\[
\frac{1}{20} = \frac{21}{420}
\]

\[
\frac{1}{30} = \frac{14}{420}
\]

\[
\frac{1}{42} = \frac{10}{420}
\]

Now, we add these fractions:

\[
\frac{70}{420} + \frac{35}{420} + \frac{21}{420} + \frac{14}{420} + \frac{10}{420} = \frac{70 + 35 + 21 + 14 + 10}{420} = \frac{150}{420}
\]

Simplify the fraction \( \frac{150}{420} \):

\[
\frac{150}{420} = \frac{15}{42} = \frac{5}{14}
\]

So, the sum of the series \( \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \frac{1}{42} \) is \( \frac{5}{14} \).