Problem 724

Drone Delivery

A depot uses $n$ drones to disperse packages containing essential supplies along a long straight road.
Initially all drones are stationary, loaded with a supply package.
Every second, the depot selects a drone at random and sends it this instruction:

If you are stationary, start moving at one centimetre per second along the road.
If you are moving, increase your speed by one centimetre per second along the road without changing direction.

The road is wide enough that drones can overtake one another without risk of collision.

Eventually, there will only be one drone left at the depot waiting to receive its first instruction. As soon as that drone has flown one centimetre along the road, all drones drop their packages and return to the depot.

Let $E (n)$ be the expected distance in centimetres from the depot that the supply packages land.
For example, $E (2) = \frac{7}{2}$ , $E (5) = \frac{12019}{720}$ , and $E (100) \approx 1427.193470$ .

Find $E (10^{8})$ . Give your answer rounded to the nearest integer.

无人机快递

一家仓库使用 $n$ 架无人机来沿着一条长直马路分发装有必需供给的包裹。
一开始所有的无人机都处于静止状态，各自装有一个供给包裹。
在每一秒，随机一架无人机会被选中并收到如下的指示：

如果无人机处于静止状态，则开始沿着马路以每秒一厘米的速度前进；
如果无人机处于运动状态，则在不改变运动方向的基础上每秒速度提高一厘米。

假设马路足够宽，因此无人机在加速后可以超过其它无人机而不会发生碰撞。

最终，仓库中只剩下一台无人机从未收到过指令。当这台无人机首次收到指令并沿着马路恰好前进了一厘米时，所有无人机立即在原地放下包裹，并返回仓库。

记 $E (n)$ 为供给包裹落点距离仓库的期望距离，以厘米计。
例如， $E (2) = \frac{7}{2}$ ， $E (5) = \frac{12019}{720}$ ， $E (100) \approx 1427.193470$ 。

求 $E (10^{8})$ ，并将你的答案四舍五入取整。