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Understanding series convergence is essential when dealing with infinite series. In general, a series converges when the partial sums approach a specific value as the number of terms increases. For alternating series—series in which the terms alternate in sign—convergence can be assessed using specific tests. The Alternating Series Test is a popular method that is particularly effective for sequences of the form \(\sum (-1)^{n} b_n\), where \(b_n\) are positive sequences.There are two primary conditions to show convergence of alternating series:

• The sequence \(b_n\), excluding the alternating signs, must be decreasing.
• The limit of \(b_n\) as \(n\) approaches infinity must be zero.

If both conditions are satisfied, the alternating series converges. This test not only helps simplify the process of determining convergence but also ensures a methodical approach.

A decreasing sequence is one where each term is less than or equal to the preceding term. In mathematical terms, a sequence \(a_n\) is decreasing if \(a_{n+1} \leq a_n\) for all \(n\). This is crucial for the Alternating Series Test.In the given problem, we identified the sequence \(a_n = \frac{1}{e^n}\). To demonstrate that this sequence is decreasing, we compare consecutive terms.Let's observe:

• \(a_{n+1} = \frac{1}{e^{n+1}}\)
• \(a_n = \frac{1}{e^n}\)

Because \(e^{n+1} > e^n\), naturally, \(\frac{1}{e^{n+1}} < \frac{1}{e^n}\). Thus, \(a_{n+1} < a_n\), confirming that the sequence is indeed decreasing.This decrease is pivotal as it fulfills one of the crucial conditions needed to confirm the convergence of an alternating series.

The limit condition is a critical part of using the Alternating Series Test. It ensures that as the sequence progresses, its terms tend closer and closer to zero. Specifically, for a series to converge via this test, the sequence \(a_n\) must satisfy:\[ \lim_{n \to \infty} a_n = 0 \]In our example, we have:

• \(a_n = \frac{1}{e^n}\)

As \(n\) increases, \(e^n\) grows exponentially, thus making \(\frac{1}{e^n}\) shrink towards zero.Mathematically, this is observed by:\[ \lim_{n \to \infty} \frac{1}{e^n} = 0 \]When this limit condition is satisfied, combined with the sequence being decreasing, it provides solid proof through the Alternating Series Test that the series converges.Remember, the convergence hinges on both the decreasing nature of the sequence and the limit condition being met. These steps ensure the series' terms are diminishing appropriately as they alternate.