Problem 39 - Essential Calculus

Question

Find the sum of the convergent series. $$ 1+0.1+0.01+0.001+\cdots $$

Step-by-Step Solution

Answer

The sum of the series is 1.1111

Step 1: Identify the first term and the common ratio

The first term \(a\) is 1 and the common ratio \(r\) is 0.1

Step 2: Apply the formula for the sum of a convergent geometric series

Using the formula \(S = \frac{a}{1-r}\), we substitute \(a = 1\) and \(r = 0.1\) into the formula. Thus the sum \(S\) becomes \(S = \frac{1}{1-0.1} = \frac{1}{0.9}\)

Step 3: Simplify the expression

By simplifying the expression we get the sum of the series as \(S = \frac{1}{0.9} = 1.1111\)

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. In our original exercise, we have the series:

• First term: 1
• Second term: 0.1 (which is 1 multiplied by 0.1)
• Third term: 0.01 (which is 0.1 multiplied by 0.1)

This pattern continues indefinitely with each term getting progressively smaller. The term "geometric" refers to the method of index advancement by a constant factor (the common ratio). Geometric series can either converge or diverge depending on the value of the common ratio. If the absolute value of the common ratio is less than one (which is the case in our example), the series converges.

Series Sum Formula

The sum of an infinite geometric series can be calculated using the series sum formula, which is applicable only if the series converges. The formula is given by:\[S = \frac{a}{1-r}\]where:

• \(S\) is the sum of the series.
• \(a\) is the first term of the series.
• \(r\) is the common ratio.

For our series in the example

• \(a = 1\)
• \(r = 0.1\)

Plugging these values into the formula, the sum becomes \[S = \frac{1}{1-0.1} = \frac{1}{0.9}\]This formula sums the infinite series, capitalizing on the fact that the terms decrease and approach zero as the series progresses, ensuring a finite sum.

Common Ratio

The common ratio is a crucial part of understanding a geometric series. It is essentially the factor by which we multiply each term to get to the next term.In the context of our series, the common ratio \(r\) is 0.1. What makes the series converge is that this common ratio is less than one.

• This ensures that each term in the series is smaller than the previous one.
• It also guarantees that as we sum an infinite number of terms, the total sum reaches a finite value.

Understanding the common ratio helps identify the behavior and sum of the series. A common ratio less than one results in the terms diminishing over time, while a ratio greater than or equal to one would cause the series to diverge or not have a finite sum.