Geometric Series Sum Calculation For 7/9 + 7/27 + 7/81

In the realm of mathematics, series play a crucial role in understanding various phenomena, from the behavior of sequences to the modeling of physical systems. Among the different types of series, geometric series hold a special place due to their elegant properties and wide range of applications. This article delves into the fascinating world of geometric series, focusing on the specific series 7/9 + 7/27 + 7/81 + 7/243 + ... We will explore the fundamental concepts behind geometric series, derive the formula for their sum, and apply this knowledge to determine the expression that defines Sn for the given series. Furthermore, we will discuss the conditions for convergence and divergence of geometric series, providing a comprehensive understanding of their behavior. This exploration will not only enhance your mathematical toolkit but also showcase the beauty and power of mathematical analysis in unraveling complex patterns and relationships.

H2: Exploring the Geometric Series 7/9 + 7/27 + 7/81 + ...

H3: Identifying the Pattern

To begin our exploration, let's closely examine the given series: 7/9 + 7/27 + 7/81 + 7/243 + .... The most important aspect of this series is recognizing that it is a geometric series. The key characteristic of a geometric series is that each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. By observing the series, we can identify that the first term (a) is 7/9. To find the common ratio (r), we can divide any term by its preceding term. For instance, dividing the second term (7/27) by the first term (7/9) gives us (7/27) / (7/9) = (7/27) * (9/7) = 1/3. Similarly, dividing the third term (7/81) by the second term (7/27) also yields 1/3. This consistent ratio confirms that the series is indeed geometric, with a common ratio of 1/3. The ability to recognize and extract these fundamental components – the first term and the common ratio – is paramount in analyzing and manipulating geometric series. Understanding the structure of a geometric series allows us to predict subsequent terms, calculate partial sums, and, most importantly, determine whether the series converges to a finite value or diverges infinitely. This foundational knowledge paves the way for deeper analysis and applications of geometric series in various mathematical and real-world contexts.

H3: Deriving the Formula for the Sum of a Geometric Series

Now that we have identified our series as geometric with a first term (a = 7/9) and a common ratio (r = 1/3), we can delve into deriving the formula for the sum of its first n terms, denoted as Sn. The formula for the sum of the first n terms of a geometric series is a cornerstone concept and is derived through an elegant algebraic manipulation. Let's begin by writing out the sum Sn: Sn = a + ar + ar^2 + ar^3 + ... + ar^(n-1). To derive a compact formula, we multiply both sides of this equation by the common ratio, r: rSn = ar + ar^2 + ar^3 + ... + ar^(n-1) + ar^n. Now, we subtract the second equation from the first: Sn - rSn = (a + ar + ar^2 + ... + ar^(n-1)) - (ar + ar^2 + ar^3 + ... + ar^(n-1) + ar^n). Notice that most of the terms on the right-hand side cancel out, leaving us with: Sn - rSn = a - ar^n. We can factor out Sn on the left side: Sn(1 - r) = a - ar^n. Finally, dividing both sides by (1 - r), we obtain the formula for the sum of the first n terms of a geometric series: Sn = a(1 - r^n) / (1 - r), provided that r ≠ 1. This formula is a powerful tool, allowing us to calculate the sum of any finite geometric series given its first term, common ratio, and number of terms. In the context of our specific series, this formula will be instrumental in determining the behavior of the sum as n approaches infinity.

H3: Calculating the Sum to Infinity

While the formula for Sn gives us the sum of the first n terms, a particularly interesting question arises: what happens to the sum as n approaches infinity? In other words, can we define a sum for an infinite geometric series? The answer lies in the behavior of the common ratio, r. If the absolute value of r is less than 1 (|r| < 1), the series converges, meaning that the sum approaches a finite value as n tends to infinity. This is because the term r^n approaches 0 as n becomes very large. Conversely, if |r| ≥ 1, the series diverges, meaning that the sum either grows infinitely large or oscillates without approaching a specific value. In our case, the common ratio r is 1/3, which satisfies the condition |r| < 1. Therefore, the series converges. To find the sum to infinity, we take the limit of Sn as n approaches infinity: S = lim (n→∞) Sn = lim (n→∞) a(1 - r^n) / (1 - r). Since r^n approaches 0 as n approaches infinity when |r| < 1, the formula simplifies to: S = a / (1 - r). Plugging in our values for a (7/9) and r (1/3), we get: S = (7/9) / (1 - 1/3) = (7/9) / (2/3) = (7/9) * (3/2) = 7/6. This result demonstrates that the sum of the infinite geometric series 7/9 + 7/27 + 7/81 + ... converges to 7/6. The concept of convergence is crucial in many areas of mathematics and physics, allowing us to model and understand phenomena involving infinite processes.

H2: Determining the Expression for Sn

H3: Applying the Formula

Having established the formula for the sum of the first n terms of a geometric series, Sn = a(1 - r^n) / (1 - r), and the sum to infinity, S = a / (1 - r), we can now focus on determining the specific expression for Sn in the context of our series 7/9 + 7/27 + 7/81 + ... Recall that we identified the first term, a, as 7/9 and the common ratio, r, as 1/3. Substituting these values into the formula for Sn, we get: Sn = (7/9)(1 - (1/3)^n) / (1 - 1/3). To simplify this expression, we can first address the denominator: 1 - 1/3 = 2/3. Thus, the equation becomes: Sn = (7/9)(1 - (1/3)^n) / (2/3). Dividing by a fraction is the same as multiplying by its reciprocal, so we have: Sn = (7/9)(1 - (1/3)^n) * (3/2). Now we can simplify by canceling out a factor of 3: Sn = (7/3)(1 - (1/3)^n) * (1/2). Finally, multiplying the constants, we arrive at: Sn = (7/6)(1 - (1/3)^n). This expression provides a clear and concise representation of the sum of the first n terms of the given geometric series. It allows us to calculate the sum for any value of n, providing a valuable tool for further analysis and application of the series.

H3: Evaluating the Given Options

Now, let's consider the given options and determine which expression correctly defines Sn:

• A. lim (n → ∞) 7(1/3)^n
• B. lim (n → ∞) (7/3)(1/3)^n

These options involve limits as n approaches infinity, which is relevant to the sum to infinity (S) rather than the sum of the first n terms (Sn). Option A, lim (n → ∞) 7(1/3)^n, evaluates to 0 because (1/3)^n approaches 0 as n approaches infinity. Option B, lim (n → ∞) (7/3)(1/3)^n, also evaluates to 0 for the same reason. However, these limits represent the behavior of a single term in the series as n grows large, not the sum of the terms. Our derived expression, Sn = (7/6)(1 - (1/3)^n), represents the sum of the first n terms. As n approaches infinity, this expression approaches the sum to infinity, which we calculated as 7/6. Neither of the given options directly represents Sn. They focus on the limit of a term, not the sum. To correctly express Sn, we need the formula that captures the cumulative sum up to n terms, which is Sn = (7/6)(1 - (1/3)^n). This detailed analysis highlights the importance of understanding the distinction between the sum of a series and the behavior of individual terms as n approaches infinity.

H3: Correcting the Expressions

To accurately represent Sn using a limit, we should consider the limit of the sum Sn as n approaches infinity. We have already established that Sn = (7/6)(1 - (1/3)^n). Taking the limit as n approaches infinity, we get: lim (n → ∞) Sn = lim (n → ∞) (7/6)(1 - (1/3)^n). Since (1/3)^n approaches 0 as n approaches infinity, the expression simplifies to: lim (n → ∞) Sn = (7/6)(1 - 0) = 7/6. This limit represents the sum to infinity, which is the value that the series converges to. However, this is not the expression for Sn itself, but rather the limit of Sn. The correct expression for Sn, as we derived earlier, is Sn = (7/6)(1 - (1/3)^n). This formula provides the sum of the first n terms for any positive integer n. Understanding the difference between Sn and its limit as n approaches infinity is crucial for a complete understanding of geometric series. Sn represents a partial sum, while the limit represents the total sum if the series converges. In the context of the given options, neither option accurately represents Sn, as they both evaluate to 0 and do not capture the cumulative sum of the series. Therefore, the correct representation remains Sn = (7/6)(1 - (1/3)^n).

H2: Conclusion

In conclusion, we have thoroughly examined the geometric series 7/9 + 7/27 + 7/81 + 7/243 + ..., from identifying its pattern and deriving the formula for the sum of its first n terms to calculating the sum to infinity. We determined that the series is geometric with a first term of 7/9 and a common ratio of 1/3. We derived the formula for Sn as Sn = (7/6)(1 - (1/3)^n), which accurately represents the sum of the first n terms. Furthermore, we calculated the sum to infinity as 7/6, demonstrating the convergence of the series. Through our analysis, we clarified that the given options, lim (n → ∞) 7(1/3)^n and lim (n → ∞) (7/3)(1/3)^n, do not represent Sn but rather approach 0, as they focus on the limit of individual terms. The correct expression for Sn remains Sn = (7/6)(1 - (1/3)^n), which captures the cumulative sum of the series up to n terms. This exploration highlights the importance of understanding the fundamental concepts of geometric series, including the formula for Sn, the conditions for convergence, and the distinction between partial sums and the sum to infinity. The principles discussed here are applicable to a wide range of mathematical and real-world problems, solidifying the significance of geometric series in mathematical analysis and its applications.