Some copying machines will make reduced copies. Suppose you copy a page 8 inches wide and it comes out \(\frac{3}{4}\) as wide. Then you copy this, and so on indefinitely. How far will the original and all the copies reach if you lay them out side by side on a long table?

An infinite geometric series is a series of numbers that continues indefinitely with each term being a fixed multiple of the previous term. This fixed multiple is called the common ratio. For example, in this exercise, we have a sequence where each copy's width is \(\frac{3}{4}\) of the previous one. Our sequence starts with the original width of 8 inches, followed by \(\frac{3}{4}\times8 = 6\) inches, then \(\frac{3}{4}\times6 = 4.5\) inches, and so on.
To understand how the widths add up, we must recognize that we're dealing with an infinite geometric series. The beauty is, when the common ratio \(r\) is between -1 and 1 (excluding -1 and 1), the series converges to a finite sum. This means that even though the series is infinite, it sums up to a specific number.
In this example, the series converges even though it keeps going indefinitely.

The common ratio in a geometric series is the factor by which we multiply each term to get the next term. In this exercise, the common ratio \(r\) is \(\frac{3}{4}\).
This means every term is \(\frac{3}{4}\) times the previous term. For instance, we start with 8 inches, and the first copy is \( \frac{3}{4}\times8 = 6 \) inches.
The second copy is \( \frac{3}{4}\times6 = 4.5 \) inches, and this pattern continues indefinitely. By knowing the common ratio, we can easily calculate subsequent terms in our geometric series.
It's essential to ensure the ratio is less than 1 in absolute value for the series to converge.
In applications, always check the common ratio to understand how quickly the terms decrease (or increase) in the series.

Finding the sum of an infinite geometric series involves a handy formula when you know the first term \(a\) and the common ratio \(r\). The formula is:
\[ S = \frac{a}{1 - r} \]
In this exercise, the first term \(a\) is 8 inches, and the common ratio \(r\) is \(\frac{3}{4}\). Plugging in these values, we get:
\[ S = \frac{8}{1 - \frac{3}{4}} = \frac{8}{\frac{1}{4}} = 32 \] inches.
This means if we keep making smaller and smaller copies indefinitely and lay them all side by side, their total width will be 32 inches. This calculation simplifies a potentially overwhelming task to just two known values: the first term and the common ratio.

Series convergence pertains to whether the sum of a series approaches a specific value as more terms are added. For a geometric series to converge, the absolute value of the common ratio \(r\) must be less than 1.
In our exercise, the common ratio is \(\frac{3}{4}\), which meets this criterion as \(\frac{3}{4} < 1\). Hence the series converges.
Mathematically, the infinite sum won't keep growing indefinitely but rather approach the value calculated using the sum formula. This is highly practical as it means lengthy calculations involving infinite terms are unnecessary.
Understanding convergence helps in solving real-world problems where infinite series models naturally occurring phenomena. It's key in verifying that the total extent—like the total width in this problem—can be measured and is finite.