Problem 825

Chasing Game

Two cars are on a circular track of total length $2n$, facing the same direction, initially distance $n$ apart.

They move in turn. At each turn, the moving car will advance a distance of $1$, $2$ or $3$, with equal probabilities.

The chase ends when the moving car reaches or goes beyond the position of the other car. The moving car is declared the winner.

Let $S(n)$ be the difference between the winning probabilities of the two cars.

For example, when $n = 2$, the winning probabilities of the two cars are $\frac 9 {11}$ and $\frac 2 {11}$, and thus $S(2) = \frac 7 {11}$.

Let $\displaystyle T(N) = \sum_{n = 2}^N S(n)$.

You are given that $T(10) = 2.38235282$ rounded to $8$ digits after the decimal point.

Find $T(10^{14})$, rounded to $8$ digits after the decimal point.

追逐游戏

在长度为$2n$的环形赛道上有两辆汽车沿相同方向前进，初始距离为$n$。

两辆汽车轮流行动，每次行动时会以相等概率随机向前移动距离$1$、$2$或$3$。

当其中一辆汽车追上或超过另一辆汽车时，游戏结束，前者获胜。

记$S(n)$为两车获胜概率之差。

例如，当$n = 2$时，两车获胜的概率分别是$\frac 9 {11}$和$\frac 2 {11}$，因此$S(2) = \frac 7 {11}$。

记$\displaystyle T(N) = \sum_{n = 2}^N S(n)$。

已知$T(10) = 2.38235282$，四舍五入至小数点后$8$位。

求$T(10^{14})$，并将你的答案四舍五入至小数点后$8$位。