Find The Sum Of The Infinite Geometric Series 1 + 1/3 + 1/9 + 1/27 + ...
This article delves into the fascinating world of infinite geometric series, specifically addressing the question of finding the sum of the series 1 + 1/3 + 1/9 + 1/27 + .... We will explore the fundamental concepts behind geometric series, derive the formula for calculating the sum of an infinite geometric series, and apply this formula to solve the given problem. Furthermore, we will discuss the conditions under which an infinite geometric series converges and diverges, providing a comprehensive understanding of this important mathematical concept.
Understanding Geometric Series
At the heart of our problem lies the concept of a geometric series. A geometric series is a series where each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. In the series 1 + 1/3 + 1/9 + 1/27 + ..., we can observe that each term is obtained by multiplying the previous term by 1/3. Therefore, this is a geometric series with a common ratio of 1/3. Understanding the properties of geometric series is crucial for determining their sums, especially when dealing with infinite series. The general form of a geometric series is a + ar + ar^2 + ar^3 + ..., where 'a' is the first term and 'r' is the common ratio. The behavior of a geometric series, whether it converges to a finite sum or diverges to infinity, depends heavily on the value of the common ratio 'r'. Geometric series have wide applications in various fields, including finance, physics, and computer science, making their study essential in mathematics.
Identifying the First Term and Common Ratio
To effectively work with geometric series, it's essential to accurately identify the first term (a) and the common ratio (r). In the given series, 1 + 1/3 + 1/9 + 1/27 + ..., the first term, a, is clearly 1. The common ratio, r, can be found by dividing any term by its preceding term. For instance, dividing the second term (1/3) by the first term (1) gives us 1/3. Similarly, dividing the third term (1/9) by the second term (1/3) also yields 1/3. This confirms that the common ratio, r, is 1/3. Correctly identifying a and r is a foundational step in applying the formula for the sum of a geometric series. These values dictate the series' behavior and ultimate sum, making their precise determination critical for solving problems related to geometric series. Understanding how to extract these key parameters is fundamental to analyzing and manipulating geometric sequences and series effectively.
Convergence and Divergence of Geometric Series
A crucial aspect of dealing with infinite geometric series is understanding the concepts of convergence and divergence. An infinite geometric series converges if its sum approaches a finite value as the number of terms increases indefinitely. Conversely, it diverges if the sum grows without bound. The convergence or divergence of a geometric series is determined by the absolute value of the common ratio, |r|. Specifically, an infinite geometric series converges if |r| < 1 and diverges if |r| ≥ 1. In the given series, 1 + 1/3 + 1/9 + 1/27 + ..., the common ratio r is 1/3. Since |1/3| < 1, this series converges. Understanding the conditions for convergence is vital because the formula for the sum of an infinite geometric series is only applicable when the series converges. Ignoring this condition can lead to incorrect results. The convergence criterion is a fundamental concept in the analysis of infinite series and has significant implications in various mathematical and scientific applications.
The Formula for the Sum of an Infinite Geometric Series
When an infinite geometric series converges (i.e., |r| < 1), we can calculate its sum using a specific formula. This formula provides a direct way to find the finite value that the series approaches as the number of terms goes to infinity. The formula for the sum (S) of an infinite geometric series is given by:
S = a / (1 - r)
where a is the first term and r is the common ratio. This formula is a cornerstone in the study of geometric series and is derived using limit concepts. It's essential to remember that this formula is only valid when |r| < 1, as it's based on the series converging to a finite sum. The derivation of this formula involves manipulating the partial sums of the geometric series and taking the limit as the number of terms approaches infinity. The formula's simplicity belies its power in solving problems involving infinite sums, making it an indispensable tool in mathematics and related fields. Its application allows for the efficient calculation of sums that would otherwise be difficult or impossible to determine directly.
Deriving the Formula
The derivation of the formula S = a / (1 - r) for the sum of an infinite geometric series is a beautiful application of algebraic manipulation and limit concepts. Let's consider the partial sum of the first n terms of the geometric series, denoted by S_n:
S_n = a + ar + ar^2 + ... + ar^(n-1)
Now, multiply both sides of the equation by the common ratio r:
rS_n = ar + ar^2 + ar^3 + ... + ar^n
Next, subtract the second equation from the first:
S_n - rS_n = a - ar^n
Factor out S_n on the left side:
S_n(1 - r) = a - ar^n
Divide both sides by (1 - r):
S_n = (a - ar^n) / (1 - r)
Now, as n approaches infinity, if |r| < 1, then r^n approaches 0. Therefore, the limit of S_n as n approaches infinity is:
S = lim (n→∞) S_n = a / (1 - r)
This derivation clearly shows how the formula arises from the fundamental properties of geometric series and the concept of limits. Understanding this derivation provides a deeper appreciation for the formula and its applicability.
Applying the Formula to the Problem
Now, let's apply the formula S = a / (1 - r) to solve the given problem: 1 + 1/3 + 1/9 + 1/27 + .... We have already identified the first term a as 1 and the common ratio r as 1/3. Since |1/3| < 1, the series converges, and we can use the formula to find the sum.
Substitute the values of a and r into the formula:
S = 1 / (1 - 1/3)
Simplify the denominator:
S = 1 / (2/3)
Divide 1 by 2/3 (which is equivalent to multiplying by 3/2):
S = 1 * (3/2)
S = 3/2
Therefore, the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ... is 3/2. This straightforward application of the formula demonstrates its efficiency in calculating the sum of a convergent geometric series. The process involves identifying the key parameters and plugging them into the formula, making it a readily accessible tool for solving such problems.
Analyzing the Options
We have determined that the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ... is 3/2. Now, let's analyze the given options to identify the correct answer:
(1) 4/3 (2) 3/2 (3) 5/2 (4) 13/9
Comparing our calculated sum (3/2) with the options, we can clearly see that option (2) matches our result. Therefore, the correct answer is 3/2. This step is crucial in problem-solving to ensure that the calculated answer aligns with the available choices. It also serves as a final check to confirm the accuracy of the solution. In this case, the analysis confirms that our application of the formula and subsequent calculations were correct, leading us to the right answer.
Conclusion
In conclusion, we have successfully determined the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ... to be 3/2. This was achieved by understanding the concept of geometric series, identifying the first term and common ratio, applying the formula for the sum of an infinite geometric series, and verifying the result against the given options. This problem exemplifies the power and elegance of mathematical formulas in solving complex problems. The study of geometric series is fundamental in mathematics, with applications extending to various fields. Understanding the conditions for convergence and the formula for the sum of an infinite geometric series provides a valuable tool for problem-solving and mathematical analysis.