Sum of Infinite Series

The given series can be rewritten as: $\sum_{n=1}^{\infty} \left(\frac{1}{5}\right)^n \left[\left(\frac{1}{5}\right)^n - \left(\frac{1}{7}\right)^n\right] = \sum_{n=1}^{\infty} \left[\left(\frac{1}{25}\right)^n - \left(\frac{1}{35}\right)^n\right]$

This splits into two infinite geometric series: $S_1 = \sum_{n=1}^{\infty} \left(\frac{1}{25}\right)^n = \frac{1/25}{1 - 1/25} = \frac{1}{24}$ $S_2 = \sum_{n=1}^{\infty} \left(\frac{1}{35}\right)^n = \frac{1/35}{1 - 1/35} = \frac{1}{34}$

Sum $= S_1 - S_2 = \frac{1}{24} - \frac{1}{34} = \frac{34 - 24}{816} = \frac{10}{816} = \frac{5}{408}$.