Problem 3 - Calculus: Early Transcendentals

Question

Calculate the sum of the series \( \sum_{n = 1}^{\infty} a_n \) whose partial sums are given. \( s_n = 2 - 3(0.8)^n \)

Step-by-Step Solution

Answer

The sum of the series is 2.

Step 1: Understanding the Series and Partial Sums

The series in question is defined by its partial sums: \( s_n = 2 - 3(0.8)^n \). This means that the nth partial sum of the series \( \sum_{n=1}^{\infty} a_n \) is given by this expression. We need to find the sum of the series, which is the limit of these partial sums as \( n \) approaches infinity.

Step 2: Finding the Limit of the Partial Sums

To find the sum of the series, compute \( \lim_{n \to \infty} s_n \). Given \( s_n = 2 - 3(0.8)^n \), as \( n \) approaches infinity, \( (0.8)^n \) approaches zero because \( 0 < 0.8 < 1 \). Thus, \( \lim_{n \to \infty} s_n = 2 - 3 \times 0 = 2 \).

Step 3: Conclusion on the Sum of the Series

Since the limit of the partial sums \( s_n \) as \( n \to \infty \) is 2, the sum of the infinite series \( \sum_{n=1}^{\infty} a_n \) is equal to this limit. Therefore, the sum is 2.

Key Concepts

Understanding Partial Sums

In the context of infinite series, partial sums are significant because they represent the sum of the first "n" terms of the series. Imagine you have a series represented as \( \sum_{n=1}^{\infty} a_n \). The partial sum \( s_n \) would be the sum for the first "n" terms, specifically:

• \( s_1 = a_1 \)
• \( s_2 = a_1 + a_2 \)
• \( s_3 = a_1 + a_2 + a_3 \)
• ... and so on.

Let's relate this to our original problem: the given series is defined by its partial sums: \( s_n = 2 - 3(0.8)^n \). This expression calculates each partial sum by taking 2 and subtracting it by \(3(0.8)^n\). As "n" increases, the term \( (0.8)^n \) decreases, influencing the overall value of the partial sum calculated for each stage. This is essential in determining how the series changes over its infinite nature.

Limit of a Sequence

The limit of a sequence is a crucial concept when dealing with infinite series. It tells us what value the sequence of partial sums approaches as "n" becomes very large. Consider our problem where the partial sums sequence is defined as \( s_n = 2 - 3(0.8)^n \). Observing this function, we ponder on how it behaves as "n" approaches infinity. The term \( (0.8)^n \), being a fraction less than one, decreases exponentially and heads towards zero as "n" becomes larger. Thus, when computing the limit of \( s_n = 2 - 3(0.8)^n \), the term \( 3(0.8)^n \) eventually contributes nothing as its value tends towards zero:

• \( \lim_{n \to \infty} s_n = 2 - 3 \times 0 \)
• The limit is \( 2 \).

This tells us the value that the sequence of partial sums progressively approaches over an infinite number of terms. This value is central to our understanding of the series as a whole.

Convergence of Series

Convergence in the context of series implies that as "n" approaches infinity, the sequence of partial sums settles towards a single, finite value. For a series \( \sum_{n=1}^{\infty} a_n \) to converge, there must exist a limit for its partial sums \( s_n \). If this limit is finite, the series is said to "converge"; otherwise, it "diverges."In our exercise, the partial sums \( s_n = 2 - 3(0.8)^n \) converge to a limit of \( 2 \) as "n" approaches infinity. This indicates the series converges, with the sum of the series being exactly this limit. Why is convergence important? Knowing a series converges and knowing its total sum are separate things. Convergence tells us about the behavior of the series — that no matter how many terms we add, the sequence won't explode to infinity or wobble indefinitely. Its sum remains steady at 2, giving us insight into both the series' character and its practicality in applications.