# Solution to "I Would Walk Five Hundred Miles"

In the first quatrain I would walk

Then I would walk 500 miles for half of the additional miles. In other words I would walk $500\left( \frac{500^2}{2}\right)=\frac{500^3}{2}$ miles more, for a total of

Then I would walk 500 miles for a third of the last additional miles (500 times 1/3 of the last term). In other words, $500\frac{1}{3}\left(\frac{500^3}{2}\right)=\frac{500^4}{2\cdot 3}$ miles more, for a total of

Continuing this pattern an infinite number of times, I would walk

For any value $x$, the exact Maclaurin series expansion of $e^x$ is

Then, to put the number of miles I would walk into this form, we will factor out one of the $500$'s. So

Then I would walk $500e^{500}\approx 7.018\times 10^{219}$ miles.

This distance is $1.53 \times 10^{196}$ times larger than the lower bound for the estimated diameter of the universe.