1. The problem asks to find the sum of the series $ \sum_{j=15}^{51} \frac{1}{j} $ with an increment of 2 in the index. 2. This means we sum the terms $ \frac{1}{15} + \frac{1}{17} + \frac{1}{19} + \cdots + \frac{1}{51} $. 3. The formula for the sum of terms with a step is $ \sum_{k=0}^n \frac{1}{a + kd} $, where $a=15$, $d=2$, and $k$ runs until $a + kd \leq 51$. 4. Calculate the number of terms: $ n = \frac{51 - 15}{2} = 18 $, so there are 19 terms (from $k=0$ to $k=18$). 5. Write out the sum explicitly: $$ S = \sum_{k=0}^{18} \frac{1}{15 + 2k} = \frac{1}{15} + \frac{1}{17} + \frac{1}{19} + \cdots + \frac{1}{51} $$ 6. Compute the sum numerically: $$ S \approx 0.0667 + 0.0588 + 0.0526 + 0.0476 + 0.0435 + 0.04 + 0.037 + 0.0345 + 0.0323 + 0.0303 + 0.0286 + 0.027 + 0.0256 + 0.0244 + 0.0233 + 0.0222 + 0.0213 + 0.0204 + 0.0196 $$ 7. Adding these values gives approximately: $$ S \approx 0.7157 $$ Therefore, the sum $ \sum_{j=15, j\text{ step }2}^{51} \frac{1}{j} \approx 0.7157 $.