Solution to "I Would Walk Five Hundred Miles"


In the first quatrain I would walk

\underbrace{500}_{\text{500 miles}}+\underbrace{500(500)}_{\text{500 more for each of the first 500}}=500+500^2.


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

\left[500+500^2+\frac{500^3}{2}\right] \text{ miles}.


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

\left[500+500^2+\frac{500^3}{2}+\frac{500^4}{2\cdot 3}\right] \text{ miles}.


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

\left[500+500^2+\frac{500^3}{2}+\frac{500^4}{2\cdot 3}+\frac{500^5}{2\cdot 3\cdot 4}+\frac{500^6}{2\cdot 3\cdot 4\cdot 5}+\cdots\right] \text{ miles}=\sum\limits_{n=0}^{\infty}{\frac{500^{n+1}}{n!}} \text{ miles}.


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

e^x=\sum\limits_{n=0}^{\infty}{\frac{x^n}{n!}}.


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

\sum\limits_{n=0}^{\infty}{\frac{500^{n+1}}{n!}}=500\sum\limits_{n=0}^{\infty}{\frac{500^{n}}{n!}}=500e^{500}\approx 7.018 \times 10^{219}.


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.