Determining The Expression For S_n When A_n Is 5(1/3)^n

In this article, we will delve into the mathematical expression that defines the sequence S_n, given that the sequence a_n is defined as 5(1/3)^n. This involves understanding the nature of the sequence a_n, exploring the concept of limits, and determining how these elements combine to define S_n. We'll dissect the potential expressions provided, analyze their behavior as n approaches infinity, and ultimately, identify the correct definition of S_n.

The heart of our discussion lies in deciphering how a sequence converges, especially when it involves exponential decay. The sequence a_n = 5(1/3)^n is a classic example of a geometric sequence with a common ratio less than 1, leading to its convergence towards zero. The crucial aspect is to understand not just the convergence but also the rate at which it approaches its limit, which plays a pivotal role in defining S_n. Let's embark on this mathematical journey to unravel the expression for S_n.

Analyzing the Sequence a_n = 5(1/3)^n

To accurately define S_n, a thorough analysis of the sequence a_n = 5(1/3)^n is essential. This sequence is a geometric sequence, a fundamental concept in mathematics, where each term is obtained by multiplying the previous term by a constant ratio. In this case, the initial term is 5, and the common ratio is 1/3. The behavior of geometric sequences is well-defined and depends significantly on the common ratio. When the absolute value of the common ratio is less than 1, as it is here (1/3 < 1), the sequence converges to 0. This means that as n becomes larger and larger, the terms of the sequence get closer and closer to 0.

The general form of a geometric sequence is a_n = ar^(n-1) or a_n = ar^n, where a is the first term and r is the common ratio. In our case, a can be considered as 5 when the sequence is written as a_n = 5(1/3)^n. The exponent n plays a crucial role in the sequence's behavior. As n increases, (1/3)^n decreases rapidly, causing the entire term 5(1/3)^n to approach zero. This convergence is a key aspect of understanding the limit of the sequence as n approaches infinity. We can express this mathematically as:

lim (n→∞) 5(1/3)^n = 0

This limit tells us that the terms of the sequence become infinitesimally small as n grows, which is a crucial piece of information when determining the nature of S_n. To further understand the behavior of S_n, we need to consider the concept of infinite series, which involves summing the terms of a sequence. This summation can either converge to a finite value or diverge, depending on the nature of the sequence. In the case of geometric series, the convergence or divergence is determined by the common ratio. Given that our common ratio is 1/3, which is less than 1, the sum of the series will converge. This convergence is what allows us to define S_n in a meaningful way.

Exploring the Concept of Limits

Central to determining the expression for S_n is a solid understanding of limits. The concept of a limit in mathematics, particularly in calculus and analysis, describes the value that a function (or sequence) approaches as the input (or index) approaches some value. In the context of sequences, we are often interested in the limit as n, the index, approaches infinity. This limit tells us the value that the terms of the sequence get arbitrarily close to as we go further and further along in the sequence.

The formal definition of a limit involves epsilon-delta arguments, but for our purposes, an intuitive understanding is sufficient. When we say that the limit of a sequence a_n as n approaches infinity is L, written as:

lim (n→∞) a_n = L

we mean that for any small positive number (epsilon), we can find a point in the sequence beyond which all terms are within that small distance of L. In simpler terms, the sequence gets closer and closer to L as n increases without bound. In our specific case, the sequence a_n = 5(1/3)^n has a limit of 0 as n approaches infinity. This is because the term (1/3)^n becomes vanishingly small as n increases. The exponential decay caused by the fraction 1/3 raised to the power of n ensures that the sequence converges towards zero.

The concept of limits is crucial for understanding infinite series. An infinite series is the sum of the terms of an infinite sequence. The limit of the sequence of partial sums determines whether the series converges or diverges. The nth partial sum, often denoted as S_n, is the sum of the first n terms of the sequence. If the sequence of partial sums approaches a finite limit as n approaches infinity, then the infinite series converges, and that limit is the sum of the series. If the sequence of partial sums does not approach a finite limit, the series diverges. Understanding these limit behaviors is key to correctly identifying the expression for S_n.

Determining the Expression for S_n

Given that a_n = 5(1/3)^n, the determination of the expression for S_n involves understanding what S_n represents. S_n is typically defined as the sum of the first n terms of the sequence a_n. Therefore, S_n can be written as:

S_n = a_1 + a_2 + a_3 + ... + a_n

Substituting the expression for a_n, we get:

S_n = 5(1/3)^1 + 5(1/3)^2 + 5(1/3)^3 + ... + 5(1/3)^n

This is a finite geometric series. A geometric series is a series where the ratio between consecutive terms is constant. In this case, the common ratio is 1/3. The sum of a finite geometric series can be calculated using the formula:

S_n = a(1 - r^n) / (1 - r)

where a is the first term, r is the common ratio, and n is the number of terms. In our case, the first term a is 5(1/3), and the common ratio r is 1/3. Plugging these values into the formula, we get:

S_n = 5(1/3) [1 - (1/3)^n] / [1 - (1/3)]

Simplifying this expression:

S_n = (5/3) [1 - (1/3)^n] / (2/3)
S_n = (5/3) [1 - (1/3)^n] * (3/2)
S_n = (5/2) [1 - (1/3)^n]

This expression for S_n represents the sum of the first n terms of the sequence a_n. It's important to note that this is a closed-form expression, meaning it directly calculates S_n without requiring the summation of individual terms. The behavior of S_n as n approaches infinity is also of interest. As n becomes very large, (1/3)^n approaches 0, and S_n approaches 5/2. This limit represents the sum of the infinite geometric series:

lim (n→∞) S_n = lim (n→∞) (5/2) [1 - (1/3)^n] = 5/2

Analyzing Potential Expressions for S_n

Now, let's analyze the potential expressions that might define S_n. Given our understanding of the sequence a_n and the sum of geometric series, we can evaluate which expressions are plausible. The expressions presented are in the form of limits, which is fitting, as the sum of an infinite series is defined as the limit of the partial sums. The correct expression for S_n should accurately represent the sum of the series as n approaches infinity.

One expression presented is:

lim (n→∞) 5(1/3)^n

This expression represents the limit of the nth term of the sequence a_n as n approaches infinity. We've already established that this limit is 0. However, S_n represents the sum of the terms, not the limit of the individual terms. Therefore, this expression is incorrect. It describes the behavior of the terms themselves, not the cumulative sum.

Another expression is:

lim (n→∞) (5/2)(1/3)^n

This expression is also incorrect for a similar reason. It represents the limit of a term that includes (1/3)^n, which approaches 0 as n approaches infinity. This expression does not capture the cumulative sum of the series; it merely describes a term that converges to zero. The sum of the series should converge to a finite, non-zero value, as we've shown in our derivation of S_n.

The correct form for S_n should reflect the sum of the geometric series. We derived the closed-form expression for S_n as:

S_n = (5/2) [1 - (1/3)^n]

The limit of this expression as n approaches infinity is:

lim (n→∞) (5/2) [1 - (1/3)^n] = 5/2

This result indicates that the sum of the infinite geometric series converges to 5/2. The correct expression for S_n should therefore reflect this convergence to 5/2 as n approaches infinity. Expressions that converge to 0 do not represent the sum of the series but rather the behavior of individual terms in the sequence.

Conclusion

In conclusion, understanding the expression for S_n when a_n = 5(1/3)^n requires a deep dive into geometric sequences, limits, and series. We've established that a_n is a geometric sequence converging to 0 as n approaches infinity. S_n, defined as the sum of the first n terms of a_n, is a geometric series. By applying the formula for the sum of a finite geometric series and then taking the limit as n approaches infinity, we determined that S_n converges to 5/2.

Expressions that simply show a term approaching 0, such as lim (n→∞) 5(1/3)^n and lim (n→∞) (5/2)(1/3)^n, are incorrect because they represent the behavior of individual terms, not the cumulative sum. The correct expression must reflect the convergence of the series to a finite, non-zero value. This exploration highlights the importance of understanding the fundamental concepts of sequences and series in determining the behavior of mathematical expressions.

Ultimately, the analysis underscores the critical distinction between the behavior of the terms of a sequence and the behavior of the sum of those terms. While the terms a_n approach 0, the sum S_n approaches 5/2, demonstrating the power of infinite summation in converging geometric series. This understanding is crucial in various fields of mathematics and its applications, from calculus to physics and engineering.