You know the story – Achilles and the tortoise are having a race. The tortoise is given a head start of 100m. After some time, Achilles will arrive at where the tortoise was at, but the tortoise will have moved further. Question: find how long it will take for Achilles to catch up to the tortoise in seconds, and how far Achilles will travel when he reaches the tortoise in meters. Achilles runs at $10m/s$ and the tortoise $0.2m/s$. Use an infinite series.
Intuition
Obviously I would have to use infinite series to solve. As I see it, I should create two series: one for Achilles and one for the tortoise. For the turtle, it would roughly be $100 + \sum_{n=1}^{\infty}a_n$. The 100 is the 100 meter head start, and $a_n$ would be the series that represents the movement of the tortoise. For Achilles it would be similar.
Goal
The goal is to find $n$, which would depict the time in seconds, where Achilles will catch up to the tortoise. To do this, I would have to equate Achilles' series and the tortoise's series, evaluate each, and solve for n. To find the total distance traveled by Achilles, it would be substituting $n$ into the equation and evaluating the series.
Answers
So, any answers for $n$? I'm mainly confused as to how to formulate the infinite series, and how to solve for each series (which would involve seeing if it converges and such).
Best Answer
There is no reason that you have to use an infinite series. Achilles gains on the tortoise at a speed of $10-0.2 = 9.8 \frac {m}{s}$ and catches the tortoise in $\frac {100}{9.8} s$
If you really want to set this up as a series.
In 10 seconds the Achilles is at the tortoise's starting point. And the tortoise has moved 2 meters away. The new distance is 2 percent of the original distance. In $(10)(0.02)$ seconds Achilles will be an this new point, The tortoise will have move again creating a new gap $2\%$ of the previous gap.
$10\sum_\limits{i=0}^{\infty} 0.02^i$