Infinite Geometric Series

Bouncing Ball

Imagine throwing a tennis ball from a height of 1 meter. The ball will bounce, but each bounce height is only $\frac{1}{4}$ of the previous bounce height.

The bounce heights form a geometric sequence:

With the first term ($a$) being 1 and the ratio ($r$) being $\frac{1}{4}$.

Now, think about this: what is the total distance traveled by the ball from the moment it's thrown until it finally stops?

The ball moves down, then up, down again, up again, and so on, until it stops. The total distance is the sum of all downward paths and all upward paths. Since the ball keeps bouncing (although lower and lower), we are summing an infinite number of distances. This is called an Infinite Geometric Series.

When Does the Series Stop?

Logically, the ball will eventually stop bouncing, right? This happens because the bounce height gets smaller and smaller, approaching zero. In mathematics, this occurs if the absolute value of the ratio ($r$) is less than 1.

A series like this is called convergent, meaning its sum approaches a specific finite value (not infinity). In the ball example, $r = \frac{1}{4}$ , and since $-1 < \frac{1}{4} < 1$, the series is convergent. The ball will stop, and its total distance can be calculated.

If $|r| \ge 1$ (i.e., $r \ge 1$ or $r \le -1$), the bounce height won't decrease or might even increase. The series is called divergent, and its sum is infinite ($\pm \infty$).

Calculating the Sum of an Infinite Series

How do we calculate the sum of a convergent infinite series? We start with the formula for the sum of the first $n$ terms of a geometric series:

For an infinite series, we look for the value of $S_n$ as $n$ becomes very large (approaches infinity). If the series is convergent ($-1 < r < 1$ ), then the value of $r^n$ will approach 0 as $n$ approaches infinity. Example: $(\frac{1}{4})^2 = \frac{1}{16}$, $(\frac{1}{4})^3 = \frac{1}{64}$ , $(\frac{1}{4})^{10} \approx 0.00000095$. The larger $n$ gets, the closer $(\frac{1}{4})^n$ gets to 0.

So, for $n \to \infty$ and $-1 < r < 1$, we have $r^n \to 0$. The formula becomes:

Thus, the formula for the sum of a convergent infinite geometric series is:

Where: * $S_\infty$ = Sum of the infinite series * $a$ = First term * $r$ = Ratio ($-1 < r < 1$)

Calculating the Total Distance

We can calculate the total distance traveled by the ball using the $S_\infty$ formula. There are two parts to the path:

1. Downward Path: The ball falls from height $a$, then falls again after the first bounce ($ar$), falls again after the second bounce ($ar^2$), and so on.

   • Downward series: $a + ar + ar^2 + ar^3 + \dots$
   • First term ($a_d$) = $a$
   • Ratio ($r_d$) = $r$
   • Sum of downward path: $S_{\text{down}} = \frac{a_d}{1 - r_d} = \frac{a}{1 - r}$

2. Upward Path: The ball moves up after the first bounce ($ar$), up again after the second bounce ($ar^2$), and so on.

   • Upward series: $ar + ar^2 + ar^3 + \dots$
   • First term ($a_u$) = $ar$
   • Ratio ($r_u$) = $r$
   • Sum of upward path: $S_{\text{up}} = \frac{a_u}{1 - r_u} = \frac{ar}{1 - r}$

Total Distance = Sum of Downward Path + Sum of Upward Path

For the tennis ball example with $a = 1$ meter and $r = \frac{1}{4}$:

So, the total distance traveled by the ball until it stops is $\frac{5}{3}$ meters.

Alternative Method (Following the Hint): The total distance can also be calculated as: First downward path + 2 times the sum of all upward paths.

The result is the same!