Evaluate The Sum Of A Geometric Series A Step-by-Step Guide

In mathematics, evaluating series is a fundamental skill, and geometric series hold a special place due to their predictable nature. This article delves into the evaluation of a specific geometric series, providing a comprehensive, step-by-step approach to understanding the underlying principles and arriving at the solution. We will explore the formula for the sum of a geometric series and apply it to the given problem, ensuring a clear and concise explanation for readers of all backgrounds.

Understanding Geometric Series

At its core, a geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant factor. This constant factor is known as the common ratio, often denoted by 'r'. The general form of a geometric series is:

a, ar, ar^2, ar^3, ...

where 'a' represents the first term and 'r' is the common ratio. Understanding the characteristics of geometric series is crucial for successfully evaluating their sums. One key aspect is the behavior of the series based on the value of the common ratio. If the absolute value of 'r' is less than 1 (|r| < 1), the series converges, meaning its sum approaches a finite value as the number of terms increases. Conversely, if |r| ≥ 1, the series diverges, and its sum either tends to infinity or oscillates without approaching a specific value.

The sum of a finite geometric series can be calculated using a well-established formula. This formula provides a direct method to find the sum without having to manually add each term in the series, especially when dealing with a large number of terms. The formula is given by:

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

where:

• S_n is the sum of the first 'n' terms
• a is the first term
• r is the common ratio
• n is the number of terms

This formula is a cornerstone in the evaluation of geometric series, and a clear understanding of its components is essential. The first term 'a' sets the scale for the series, while the common ratio 'r' dictates how the terms grow or shrink. The number of terms 'n' determines the extent of the series being summed. By correctly identifying these parameters from a given geometric series, we can confidently apply the formula to find its sum.

Before diving into the specific problem, it's worth noting the significance of geometric series in various fields. They appear in finance for calculating compound interest, in physics for modeling radioactive decay, and in computer science for analyzing algorithms. The ability to evaluate geometric series is therefore a valuable skill with broad applications.

Problem Statement: Evaluating the Given Series

Our task is to evaluate the sum of the following geometric series:

$\sum_{n=1}^{20} 4\left(\frac{8}{9}\right)^{n-1}$

This notation represents the sum of the first 20 terms of a geometric series. The expression inside the summation, 4(8/9)^(n-1), defines the nth term of the series. To evaluate this sum, we will leverage the formula for the sum of a finite geometric series, which we discussed earlier. The key is to correctly identify the parameters: the first term (a), the common ratio (r), and the number of terms (n).

The first term (a) can be found by substituting n = 1 into the expression for the nth term. This gives us:

a = 4(8/9)^(1-1) = 4(8/9)^0 = 4(1) = 4

Therefore, the first term of the series is 4. This value sets the starting point for our summation and is a crucial component in the geometric series sum formula.

The common ratio (r) is the factor by which each term is multiplied to obtain the next term. In this case, the common ratio is evident from the expression: it is the base of the exponent, which is 8/9. So, r = 8/9. The common ratio plays a significant role in the behavior of the geometric series. Since |r| = 8/9 < 1, the series converges, meaning its sum will approach a finite value as we include more terms.

The number of terms (n) is explicitly given in the summation notation. The sum runs from n = 1 to n = 20, indicating that we are summing 20 terms. Therefore, n = 20. This parameter determines the extent of the series we are considering for summation.

Now that we have identified all the necessary parameters – a = 4, r = 8/9, and n = 20 – we are well-equipped to apply the formula for the sum of a finite geometric series. The next step involves plugging these values into the formula and performing the necessary calculations to arrive at the final answer. Understanding how to correctly identify these parameters is fundamental to successfully evaluating any geometric series.

Applying the Formula and Calculating the Sum

Now that we have identified the parameters of our geometric series, we can proceed to apply the formula for the sum of a finite geometric series. Recall the formula:

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

We have: a = 4, r = 8/9, and n = 20. Substituting these values into the formula, we get:

S_20 = 4(1 - (8/9)^20) / (1 - 8/9)

This expression represents the sum of the first 20 terms of the given geometric series. The next step is to simplify this expression and calculate the numerical value. The order of operations is crucial here: we must first evaluate the exponent, then perform the subtractions, and finally the division and multiplication.

Let's break down the calculation step by step. First, we need to calculate (8/9)^20. This can be done using a calculator or computational tool. The result is approximately 0.09477. It's important to use a sufficient number of decimal places to maintain accuracy in the final answer.

Next, we subtract this value from 1:

1 - (8/9)^20 ≈ 1 - 0.09477 = 0.90523

This represents the term (1 - r^n) in our formula. Now, we multiply this by the first term, a = 4:

4(0.90523) ≈ 3.62092

This gives us the numerator of our expression. Finally, we need to divide by (1 - r), which is (1 - 8/9):

1 - 8/9 = 1/9

So, our final calculation is:

S_20 ≈ 3.62092 / (1/9) = 3.62092 * 9 ≈ 32.58828

Therefore, the sum of the first 20 terms of the geometric series is approximately 32.58828. This result demonstrates the power of the geometric series sum formula, allowing us to efficiently calculate the sum of a series without having to manually add each term.

It's worth noting that the sum is converging towards a finite value, as expected since the common ratio is less than 1. As we include more terms in the series, the sum will get closer and closer to a limiting value. This concept of convergence is a fundamental aspect of infinite geometric series.

Final Answer and Conclusion

In conclusion, we have successfully evaluated the sum of the geometric series:

$\sum_{n=1}^{20} 4\left(\frac{8}{9}\right)^{n-1}$

By carefully applying the formula for the sum of a finite geometric series, we determined that the sum of the first 20 terms is approximately 32.58828. This process involved identifying the key parameters – the first term (a = 4), the common ratio (r = 8/9), and the number of terms (n = 20) – and substituting them into the formula:

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

The calculation required evaluating the exponential term, performing subtractions, and finally dividing to obtain the result. The step-by-step approach ensured accuracy and clarity in the solution process. This evaluation highlights the importance of understanding geometric series and their properties. The ability to recognize a geometric series and apply the appropriate formula is a valuable skill in mathematics and various other fields.

Furthermore, this exercise reinforces the concept of convergence in geometric series. Since the common ratio (8/9) is less than 1, the series converges, and its sum approaches a finite value as the number of terms increases. This is a key characteristic of geometric series that distinguishes them from divergent series, where the sum grows infinitely.

Therefore, the final answer to the given problem is:

S ≈ 32.58828

This detailed exploration of the geometric series evaluation provides a solid understanding of the principles and techniques involved. By mastering these concepts, readers can confidently tackle similar problems and appreciate the elegance and utility of geometric series in mathematics and beyond. The sum of geometric series is a fundamental concept with broad applications, and this article serves as a comprehensive guide to its evaluation.

In summary, geometric series evaluation is a crucial mathematical skill with applications in various fields. This article provided a step-by-step guide to evaluating a specific geometric series, emphasizing the importance of understanding the underlying principles and the correct application of the sum formula. We trust that this comprehensive explanation has enhanced your understanding of geometric series and their evaluation.