Problem 53 - A Graphical Approach to College Algebra

Question

Find each sum that converges. $$100+10+1+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{1000}{9} \) or approximately 111.11 (repeating).

Step 1: Identify the sequence type

The given series is \(100 + 10 + 1 + \cdots\). Look at the sequence of terms: each term is 1/10 of the previous term. This indicates that the series is a geometric series.

Step 2: Determine the first term and common ratio

For a geometric series, the first term \(a\) is \(100\), and the common ratio \(r\) can be found by dividing the second term by the first term: \(r = \frac{10}{100} = 0.1\).

Step 3: Apply the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series is \( S = \frac{a}{1-r} \) where \( |r| < 1 \). Here, \(a = 100\) and \(r = 0.1\), which satisfies the condition \(|r| < 1\).

Step 4: Calculate the sum

Substitute the values into the formula: \[ S = \frac{100}{1-0.1} = \frac{100}{0.9} = \frac{1000}{9} \approx 111.11 \] (repeating).

Step 5: Finalize the answer

The series converges, and its sum is \( \frac{1000}{9} \), which is a finite value indicating that the series has a sum.

Key Concepts

Infinite Series

An infinite series is the summation of an infinite sequence of terms. Instead of stopping at a certain point, terms continue indefinitely. These can be daunting at first because they seem to sum up forever. However, in many cases, especially with geometric series, it is possible to find their sum.
Understanding how an infinite series behaves is crucial. It relies on whether the series converges or diverges. A convergent series will have a definite sum, while a divergent one will not. In our exercise, where the series was given as \(100 + 10 + 1 + \cdots\), we notice a pattern that it extends indefinitely but maintains a specific ratio between terms. This makes it a candidate for convergence.
Identifying if a series is geometric and verifying the common ratio helps us find out if we can calculate a real sum, even if the series seems to go on forever.

Sum Convergence

Sum convergence refers to the condition under which an infinite series approaches a specific value. This concept is vital because it tells us that, potentially, the infinite series aggregates to a finite number, despite having infinite terms. For our geometric series study, convergence happens under a straightforward condition: the absolute value of the common ratio must be less than one.

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

In our series example, where the common ratio \(r = 0.1\), it fulfills \(|r| < 1\). Therefore, we can successfully compute its sum using the formula \( S = \frac{a}{1-r} \). This formula takes the first term and adjusts it by the gap left after subtracting the common ratio from one, effectively summing an otherwise infinite series. This ensures the series \(100 + 10 + 1 + \cdots\) can be written as a nice compact value of \(\frac{1000}{9}\).

Common Ratio

The common ratio is a key player in understanding geometric series. It describes the factor by which each term of the series is multiplied to get the next term. In the context of geometric sequences, it's what makes detecting patterns easy in otherwise complicated series. For any sequence identified as geometric, the common ratio can be derived by dividing a term by the one preceding it.
In our original series \(100 + 10 + 1 + \cdots\), the common ratio \(r\) is \(0.1\). This is calculated by taking the subsequent term \(10\) and dividing it by the prior term \(100\) (\(r = \frac{10}{100}\)).
A study of geometric series centers around this ratio. If it's less than one, as we've seen, it guides us to convergence so we can find a sum. If it's one or greater, it suggests divergence, implying the series might not have a definite sum. The common ratio fundamentally informs the behavior and sum of the series.