Identifying Equivalent Infinite Series A Detailed Comparison

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In the realm of mathematical analysis, identifying equivalent series is a fundamental skill. This involves determining whether different-looking infinite sums actually converge to the same value or exhibit the same behavior. This article delves into the intricate process of analyzing and comparing series, specifically focusing on three given series to unveil their relationships and determine if any are identical. We'll dissect each series, manipulate their forms, and employ established convergence tests to arrive at a conclusive determination. This exercise not only highlights the importance of algebraic manipulation in series analysis but also underscores the power of understanding index shifting, term rearrangement, and the application of various convergence criteria in the world of infinite sums. We will thoroughly explore the nuances of each series to ensure a comprehensive comparison, shedding light on the underlying mathematical principles at play and providing a clear understanding of their equivalence, or lack thereof. This meticulous approach will empower readers to confidently tackle similar problems involving series analysis and establish a strong foundation in the concepts of convergence and divergence.

Series Definitions

We are presented with three infinite series:

(a) $\sum_{n=6}^{\infty} n\left(\frac{5}{6}\right)^n$

(b) $\sum_{n=0}{\infty}(n+2)\left(\frac{5}{6}\right)n$

(c) $\sum_{n=2}^{\infty} n\left(\frac{5}{6}\right)^{n-2}$

Our mission is to meticulously examine these series, employing a combination of mathematical manipulation and established convergence tests, to definitively determine if any of them are equivalent. This requires a deep dive into the nuances of each series, paying close attention to their starting indices, the coefficients involved, and the overall structure of the terms. By systematically comparing these elements, we can unravel the relationships between the series and confidently identify any matches or discrepancies. The process will involve techniques such as index shifting, term rearrangement, and careful consideration of the convergence behavior of each series. This in-depth analysis will not only provide the answer but also illuminate the underlying mathematical principles that govern the behavior of infinite sums. It is a journey through the intricacies of series convergence, requiring precision, attention to detail, and a solid understanding of mathematical concepts.

Analysis and Comparison

To begin, let's manipulate series (c) to see if it can be transformed into a form similar to either (a) or (b). We can rewrite the exponent in series (c) by adding and subtracting 2 in the exponent:

∑n=2∞n(56)n−2=∑n=2∞n(56)n(56)−2=(65)2∑n=2∞n(56)n=3625∑n=2∞n(56)n\sum_{n=2}^{\infty} n\left(\frac{5}{6}\right)^{n-2} = \sum_{n=2}^{\infty} n\left(\frac{5}{6}\right)^{n}\left(\frac{5}{6}\right)^{-2} = \left(\frac{6}{5}\right)^2 \sum_{n=2}^{\infty} n\left(\frac{5}{6}\right)^{n} = \frac{36}{25} \sum_{n=2}^{\infty} n\left(\frac{5}{6}\right)^{n}

Now, let's consider series (a) and attempt to relate it to the modified form of series (c). Series (a) starts at n = 6, while the modified series (c) starts at n = 2. To compare them effectively, we need to account for the initial terms of the modified series (c) that are not present in series (a). This involves a careful examination of the terms from n = 2 to n = 5 in series (c). We must calculate these terms explicitly and then compare the remaining infinite sums. The comparison will determine if the two series are identical or if their differences lie in the initial terms. This step-by-step analysis is crucial for understanding the behavior of the series and establishing their relationship, highlighting the importance of accounting for the starting indices in series analysis. The accuracy of this comparison hinges on meticulous calculation and a clear understanding of the impact of each term on the overall sum.

To do this, we can express the modified series (c) as:

3625∑n=2∞n(56)n=3625[∑n=25n(56)n+∑n=6∞n(56)n]\frac{36}{25} \sum_{n=2}^{\infty} n\left(\frac{5}{6}\right)^{n} = \frac{36}{25} \left[ \sum_{n=2}^{5} n\left(\frac{5}{6}\right)^{n} + \sum_{n=6}^{\infty} n\left(\frac{5}{6}\right)^{n} \right]

Now we can see the second term inside the brackets is the same as series (a), so we have:

3625∑n=2∞n(56)n=3625∑n=25n(56)n+3625∑n=6∞n(56)n\frac{36}{25} \sum_{n=2}^{\infty} n\left(\frac{5}{6}\right)^{n} = \frac{36}{25} \sum_{n=2}^{5} n\left(\frac{5}{6}\right)^{n} + \frac{36}{25} \sum_{n=6}^{\infty} n\left(\frac{5}{6}\right)^{n}

Comparing this to series (a), we see that series (c) is a scaled version of series (a) plus an additional finite sum. Therefore, series (a) and (c) are not the same. This highlights a key distinction – while the infinite tails of the series may be related, the presence of the initial finite sum and the scaling factor prevent them from being identical. This observation emphasizes the importance of considering both the infinite behavior and the initial terms when comparing series. The difference in the starting indices and the coefficients plays a crucial role in determining the overall equivalence of the series. A thorough understanding of these nuances is essential for accurate series analysis and the identification of matching series.

Next, let's examine series (b). We can rewrite series (b) as:

∑n=0∞(n+2)(56)n=∑n=0∞n(56)n+2∑n=0∞(56)n\sum_{n=0}^{\infty}(n+2)\left(\frac{5}{6}\right)^n = \sum_{n=0}^{\infty}n\left(\frac{5}{6}\right)^n + 2\sum_{n=0}^{\infty}\left(\frac{5}{6}\right)^n

The second term is a geometric series with a common ratio of 5/6, which converges. The first term is similar in form to the terms in series (a) and (c). This decomposition of series (b) into two separate series provides a valuable perspective for comparison. By isolating the geometric series, we can focus on the remaining term and assess its relationship to the other series. This approach simplifies the analysis and allows for a more targeted comparison of the individual components. Understanding the behavior of each component is crucial for determining the overall convergence and equivalence of the original series. This strategic separation technique is a powerful tool in series analysis, enabling a more granular examination of complex expressions.

To compare series (b) with series (c), we need to shift the index of series (c) to start from n=0. Let m = n - 2, so n = m + 2. When n = 2, m = 0. Then series (c) becomes:

3625∑n=2∞n(56)n=3625∑m=0∞(m+2)(56)m+2=3625(56)2∑m=0∞(m+2)(56)m=∑m=0∞(m+2)(56)m\frac{36}{25} \sum_{n=2}^{\infty} n\left(\frac{5}{6}\right)^{n} = \frac{36}{25} \sum_{m=0}^{\infty} (m+2)\left(\frac{5}{6}\right)^{m+2} = \frac{36}{25} \left(\frac{5}{6}\right)^{2} \sum_{m=0}^{\infty} (m+2)\left(\frac{5}{6}\right)^{m} = \sum_{m=0}^{\infty} (m+2)\left(\frac{5}{6}\right)^{m}

Replacing m with n, we get:

∑n=0∞(n+2)(56)n\sum_{n=0}^{\infty} (n+2)\left(\frac{5}{6}\right)^{n}

This is exactly series (b). The successful index shifting and algebraic manipulation have revealed the identity between series (b) and (c). This underscores the importance of these techniques in series analysis, allowing us to transform series into comparable forms and unveil hidden relationships. The careful tracking of index changes and the application of exponent rules are crucial for ensuring the accuracy of the transformation. This process not only demonstrates the equivalence of the series but also reinforces the fundamental principles of series manipulation. The ability to effectively shift indices is a cornerstone of advanced mathematical problem-solving, enabling us to bridge seemingly disparate expressions.

Conclusion

After a thorough analysis, we have identified that series (b) and series (c) are the same. Series (a) is different from the other two due to the starting index and the scaling factor introduced in the transformation of series (c). This exercise underscores the importance of meticulous manipulation and comparison when dealing with infinite series. The process of identifying equivalent series requires a blend of algebraic skills, understanding of index shifting, and familiarity with series convergence tests. By carefully examining the structure of each series, manipulating their forms, and comparing their terms, we can confidently determine their relationships and identify matches or discrepancies. This analytical approach is not only crucial for solving specific problems but also for developing a deeper understanding of the underlying principles of series analysis. The insights gained from this exploration empower us to tackle more complex problems involving infinite sums and sequences.

Therefore, the two series that are the same are (b) and (c).

Repair Input Keyword: Which two series from the following list are identical? (a) $\sum_{n=6}^{\infty} n\left(\frac{5}{6}\right)^n$ (b) $\sum_{n=0}{\infty}(n+2)\left(\frac{5}{6}\right)n$ (c) $\sum_{n=2}^{\infty} n\left(\frac{5}{6}\right)^{n-2}$