# How do you find the sum of the infinite geometric series 215 - 86 + 34.4 - 13.76 + ...?

Use the formula $a + a r + a {r}^{2} + a {r}^{3} + \cdots = \frac{a}{1 - r}$ (for $| r | < 1$) to get $\frac{1075}{7} \approx 153.6$.
The series $215 - 86 + 34.4 - 13.76 + \cdots$ is geometric with first term $a = 215$ and common ratio $r = - \frac{86}{215} = - \frac{2}{5} = - 0.4$.
The formula above therefore gives $\frac{a}{1 - r} = \frac{215}{1 + \frac{2}{5}} = \frac{215}{\frac{7}{5}} = \frac{215 \cdot 5}{7} = \frac{1075}{7} \approx 153.6$.