Problem 6 - Applied Calculus

Question

Find the sum, if it exists. $$ 1000+1000(1.05)+1000(1.05)^{2}+\cdots $$

Step-by-Step Solution

Answer

The series diverges; the sum does not exist.

Step 1: Identify the Series

The given expression is a series: \( 1000 + 1000(1.05) + 1000(1.05)^2 + \cdots \). This is a geometric series because each term is obtained by multiplying the previous term by a constant ratio. In this case, the first term \( a = 1000 \) and the common ratio \( r = 1.05 \).

Step 2: Check for Convergence

For an infinite geometric series \( a + ar + ar^2 + \cdots \) to converge, the absolute value of the common ratio, \( |r| \), must be less than 1. Here, \( r = 1.05 \), which is greater than 1. Therefore, the series does not converge.

Step 3: Conclude the Sum

Since the ratio \( r = 1.05 \) is greater than 1, the series does not converge and hence does not have a sum. In an infinite geometric series, if \( |r| \geq 1 \), the series diverges.

Key Concepts

Convergence

Convergence is an important concept in mathematics, especially when dealing with series. An infinite series is said to converge when the sequence of its partial sums approaches a specific, finite number as you add more and more terms.
For geometric series, like the one in this exercise, convergence depends on the value of the common ratio, denoted as \( r \).

• If \( |r| < 1 \), the series converges, and it is possible to find its sum using the formula for the sum of an infinite geometric series:

\[S = \frac{a}{1 - r}\]

• If \( |r| \geq 1 \), like in the provided exercise where \( r = 1.05 \), the series does not converge. In simple terms, the terms will keep getting larger, instead of settling around a specific value.

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. You can imagine adding up an endless number of terms, trying to reach a sum. In such series, the concept of adding endlessly might seem difficult, but it's a crucial mathematical tool.
For a geometric series like the ones in our example, to understand it fully, you need to focus on its first term and the common ratio:

• The first term (denoted as \( a \)) is what you start with. In this exercise, it's \( 1000 \).
• The common ratio (denoted as \( r \)) is what you multiply by to get from one term to the next. Here, it's \( 1.05 \).

The series can be thought of visually as a sequence of dots moving ever closer to a line, or it can be calculated by closely examining the terms and their behavior.

Divergence

Divergence is the opposite of convergence. When a series diverges, it means that as you add more terms, the series does not settle around a specific number. Instead, the terms tend to grow without bounds or oscillate indefinitely.

• In the geometric series from the exercise, since the common ratio \( r = 1.05 \) is greater than 1, each term becomes progressively larger.
• This implies that the series keeps increasing and can not approach a finite sum, thereby diverging.

Understanding divergence is crucial because it tells us the limits of what we can conclude from certain kinds of series. In practical terms, divergent series lack a meaningful sum, contrasting sharply with convergent series, which offer a well-defined, approachable result.