Find the sum of series 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + ... where the term are the reciprocals of the positive integers whose only prime factors are two's and three's :

The series you are looking at consists of the reciprocals of positive integers that can be expressed as products of the prime factors 2 and 3. This means the terms in the series are of the form \( \frac{1}{2^m \cdot 3^n} \), where \( m \) and \( n \) are non-negative integers.

Identifying the Series

The series can be written as:

• 1 (when m=0, n=0)
• 1/2 (when m=1, n=0)
• 1/3 (when m=0, n=1)
• 1/4 (when m=2, n=0)
• 1/6 (when m=1, n=1)
• 1/8 (when m=3, n=0)
• 1/9 (when m=0, n=2)
• 1/12 (when m=2, n=1)
• 1/18 (when m=1, n=2)
• 1/24 (when m=3, n=1)
• 1/27 (when m=0, n=3)
• ... and so on.

Calculating the Sum

The sum of this series can be calculated using the formula for the sum of a geometric series. The series can be split into two parts:

• Sum over \( m \): \( \sum_{m=0}^{\infty} \frac{1}{2^m} = \frac{1}{1 - \frac{1}{2}} = 2 \)
• Sum over \( n \): \( \sum_{n=0}^{\infty} \frac{1}{3^n} = \frac{1}{1 - \frac{1}{3}} = \frac{3}{2} \)

Now, the total sum of the series is the product of these two sums:

Sum = 2 \times \frac{3}{2} = 3.

Final Result

The sum of the series is 3.