Infinite series.

Step 1 of 5

Halves forever

To cross a room you must first cover half of it, then half of what is left, then half of what is left again. Zeno of Elea argued that no one can take infinitely many steps, so no one can cross the room. But the steps are $\frac12 + \frac14 + \frac18 + \dots$⁠ long, and a sum like that can end in a finite number.

On the board $a_1 = 1$⁠ and $q = 0.5$⁠, and Add the terms stands at the running totals: $1$⁠, $1.5$⁠, $1.75$⁠, $1.875, \dots$⁠ Predict the sum if you keep adding forever.

Step 2 of 5

The formula for the endless sum

An endless sum is the number the running totals $S_n$⁠ close in on as $n$⁠ grows. In the formula $S_n = \frac{a_1(1 - q^n)}{1 - q}$$S_n =$$\frac{a_1(1 - q^n)}{1 - q}$⁠, when $-1 < q < 1$⁠ the power $q^n$⁠ shrinks to zero, leaving:

$S_\infty = \frac{a_1}{1 - q}$$S_\infty =$$\frac{a_1}{1 - q}$⁠

The series is said to converge. Here the first term is $a_1 = 3$⁠ and $q = 0.75$⁠. Work out the endless sum, then move Add the terms to Forever: the glass case is as tall as the sum, and the terms fill it exactly.

Step 3 of 5

Close to 1

The closer $q$⁠ is to $1$⁠, the more slowly the terms shrink, and the bigger the sum: $\frac{1}{1 - 0.5} = 2$$\frac{1}{1 - 0.5} =$$2$⁠, $\frac{1}{1 - 0.8} = 5$$\frac{1}{1 - 0.8} =$$5$⁠. Find the ratio that gives an endless sum of $10$⁠.

Step 4 of 5

When there is no total

Not every series has a sum. When $q = 1$⁠ every term equals $a_1$⁠, and $n$⁠ terms make $n a_1$⁠: the sum grows without limit. When $q > 1$⁠ it grows faster still. A rising arithmetic sequence has no endless sum either. On the board the glass case is open at the top, with an arrow: no total. Try a few values of Ratio q and Difference d, then answer.

Step 5 of 5

0.999… = 1

The decimal $0.999\ldots$⁠, with nines for ever, is a series: $0.9 + 0.09 + 0.009 + \dots$⁠, with $a_1 = 0.9$⁠ and $q = 0.1$⁠. Here the board shows four terms; the columns are already almost level. Work out the endless sum, then move Add the terms to Forever.