Problem 118 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (a) the fourth partial sum and (b) the sum of the infinite series. $$\sum_{k=1}^{\infty} \frac{4}{10^{k}}$$

Step-by-Step Solution

Answer

The fourth partial sum of the series is 0.444, and the sum of the infinite series is 0.4444.

Step 1: Identify the First Term and the Common Ratio

Observing the series \(\sum_{k=1}^{\infty} \frac{4}{10^{k}}\), the first term (a) can be found by substituting k=1 into the equation, giving a = \(\frac{4}{10^{1}} = 0.4\). The common ratio (r) is the constant multiple that each term is of the previous, which in this instance is \(\frac{1}{10} = 0.1\).

Step 2: Find the Fourth Partial Sum

The nth partial sum of a geometric series can be calculated using the formula \(S_n = a \times \frac{1 - r^n}{1 - r}\). For the fourth partial sum (n=4), substitute a = 0.4, r = 0.1, and n=4 into the formula to give \(S_4 = 0.4 \times \frac{1 - (0.1)^4}{1 - 0.1}\).

Step 3: Calculate the Sum of the Infinite Series

The sum of an infinite geometric series where the absolute value of r is less than 1 can be calculated using the formula \(S = \frac{a}{1 - r}\). Substituting a = 0.4 and r = 0.1 into the formula gives \(S = \frac{0.4}{1 - 0.1}\). The value obtained is the sum of the infinite series.

Key Concepts

Partial Sum of a Series

When dealing with sequences and series, particularly in mathematics, the concept of the partial sum is both fundamental and widely applicable. A partial sum refers to the sum of the first few individual terms of an infinite series. Mathematically, for a series comprised of an infinite number of terms, the partial sum, denoted as, for example, \(S_n\), represents the sum of the first \(n\) terms.

To find a partial sum, particularly in a geometric series, you use the formula \(S_n = a(1 - r^n) / (1 - r)\), where \(a\) is the first term and \(r\) is the common ratio between consecutive terms. For instance, if you're provided with the first term of a series and the common ratio, it allows you to compute the sum of the series up to any desired number of terms.

In the context of the given problem, the fourth partial sum \(S_4\) is calculated using the initial term 0.4 and the common ratio 0.1, resulting in a finite sum that represents the total of the first four terms.

Convergent Series

Understanding the behavior of series as they tend to infinity is a key aspect of mathematical analysis. A convergent series is one where the sum of its terms approaches a specific value as the number of terms grows to infinity. More formally, a convergent series has a finite limit, meaning that the sum of its infinite terms is a certain, calculable number.

For a geometric series to be convergent, the common ratio \(r\) must have an absolute value less than 1. The reason lies in the nature of geometric progression: if \(|r| < 1\), each subsequent term of the series becomes progressively smaller, 'shrinking' towards zero, ensuring that the series does not spiral out of bound and instead approaches a set value. Conversely, if \(|r| ≥ 1\), the series would either diverge or oscillate without approaching a certain sum.

The relevance of this definition in our exercise is profound. Since the common ratio is 0.1, the infinite series converges, and we can find its sum using the formula for the sum of a convergent geometric series, \(S = a / (1 - r)\), which gives a precise answer to the value towards which the series is heading.

Geometric Progression

Geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This progression can be represented as a series of terms: \(a, ar, ar^2, ar^3, ...\) where \(a\) is the first term and \(r\) is the common ratio not equal to 1. The bond between consecutive terms is the defining trait of this sequence.

Geometric progressions are highly versatile and appear in various fields such as music, computer science, and economics. They are useful in modeling exponential growth or decay, calculating interest, and analyzing fractal structures. Within the exercise at hand, recognizing the sequence as geometric permits the application of specific formulas to determine sums—whether that be the sum up to a finite number of terms (the partial sum) or the sum to infinity for converging series.