# How does an infinite geometric series apply to being pushed on a swing?

Assume that the initial push gives you a 3 m swing and that a swing is $\frac{6}{7}$ of its previous swing. The total horizontal distance can be expressed as the geometric series:
$3 + 3 \left(\frac{6}{7}\right) + 3 {\left(\frac{6}{7}\right)}^{2} + 3 {\left(\frac{6}{7}\right)}^{3} + \cdots = \frac{3}{1 - \frac{6}{7}} = 21$
In theory, the total horizontal distance is $21$ m.