Problem 90 - Essential Calculus

Question

Identify the two series that are the same. (a) \(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{(n-1) 2^{n-1}}\) (b) \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n 2^{n}}\) (c) \(\sum_{n=0}^{\infty} \frac{(-1)^{n+1}}{(n+1) 2^{n}}\)

Step-by-Step Solution

Answer

Series (a) and series (b) are the same when the indexing of series (a) is adjusted.

Step 1: Understanding distinction between series

Firstly, look at the three series. It's noticeable that the exponents in series (a) and (b) are the same, however, the sums start from different indexes in each of the three sums. (a) starts from 2, (b) from 1, and (c) from 0.

Step 2: Rewriting series (a)

To compare, rewrite the series so that they start from the same index. To change the starting index of series (a) to 1 (the same as series (b)), subtract 1 from the existing index in expressions, resulting in: \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n 2^{n}}\).

Step 3: Comparing series (a) and (b)

Now that the series (a) starts from the same index as series (b), compare the two: (a): \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n 2^{n}}\) and (b): \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n 2^{n}}\). They are now the same.

Step 4: Verifying series (c) is different

Although series (c) differs in the index, examine it to confirm that it is not the same as (a) and (b). This series is: \(\sum_{n=0}^{\infty} \frac{(-1)^{n+1}}{(n+1) 2^{n}}\). The exponents of the items in the expression are not the same as in series (a) and (b) and therefore, series (c) is different from (a) and (b).

Key Concepts

Convergence of Series

When dealing with mathematical series, a fundamental question is whether the series converges or diverges. To say that a series converges means that as you add more and more terms, the sum approaches a specific finite value. For an infinite series, this property might seem counterintuitive at first. But not all series grow infinitely large; some settle into a predictable pattern.

For example, an alternating series like \(\sum_{n=1}^{\text{\textinfty}} \frac{(-1)^{n+1}}{n 2^{n}}\) changes sign with each term added. This series actually converges due to the Alternating Series Test, which provides a way to prove convergence if each successive term is smaller than the one before it in absolute value, and if the limit of the terms as n approaches infinity is zero. This concept is crucial in understanding how some infinite processes can yield finite, predictable results.

Mathematical Series

A mathematical series is a sum of terms which follow a specific rule or pattern. When we write \(\sum_{n=a}^{b} f(n)\), we imply the sum of function \(f\) evaluated at integers from \(a\) to \(b\). If \(b = \infty\), it becomes an infinite series. Different types of series include geometric, where each term is a constant multiple of the previous term, or harmonic, where each term is the reciprocal of an integer.

An example of a simple geometric series is \(\sum_{n=0}^{\infty} \frac{1}{2^n}\), which converges to 2. It's important for students to understand both the concept of individual terms and the concept of summation as a whole, which combines these terms in a meaningful way.

Series Comparison

When analyzing different series, the Series Comparison Test or Limit Comparison Test can be used to determine if a series converges by comparing it to another series whose convergence is known. In our exercise, series (a) and (b) initially look different, but upon rewriting series (a) to begin at the same index as series (b), we discovered they are actually the same series.

What Does Series Comparison Tell Us?

By comparing, we can conclude the nature of the series at hand. If a series we are testing converges, and it is less than or equal to a known convergent series, then we have proof that the series tested converges. In contrast, series (c) is not the same due to a difference in exponent base despite following the alternating series pattern, showing the importance of comparing each aspect of a series.

Infinite Series

In contrast to a series with a finite number of terms, an infinite series continues indefinitely. The alternating series \(\sum_{n=1}^{\text{\textinfty}} \frac{(-1)^{n+1}}{n 2^{n}}\), for instance, has an infinite number of terms, yet its sum approaches a finite number. Learning about infinite series is beneficial for students as it introduces the concept of limits and challenges our intuition of infinity in mathematics.

Relevance of Infinite Series

Infinite series often emerge in physics and engineering when phenomena are described by infinetly recurring processes. For instance, Fourier series enable us to express complex waveforms in terms of sine and cosine functions, unveiling patterns that are otherwise not apparent. Grasping the behavior of infinite series unlocks a deeper understanding of such processes and their applications.