Question

Is the series convergent or divergent? If convergent, is it absolutely convergent? $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} $$

Short answer

The series is convergent but not absolutely convergent.

Step-by-step solution

Identify the Series Type

The given series is \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} \). This is an alternating series because of the \((-1)^n\) term, which alternates the sign of each term.

Apply the Alternating Series Test

The Alternating Series Test states that a series of the form \( \sum (-1)^n a_n \) is convergent if the sequence \( a_n \) is positive, decreasing, and the limit of \( a_n \) as \( n \to \infty \) is 0.

Check if Sequence is Positive, Decreasing and Limiting

The sequence \( a_n = \frac{1}{\sqrt{n}} \) is positive for all \( n \geq 1 \), decreases as \( n \) increases (since \( \sqrt{n} \) increases), and \( \lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0 \). Thus, the series satisfies the conditions of the Alternating Series Test.

Determine Absolute Convergence

A series is absolutely convergent if the series of absolute values \( \sum |a_n| \) is convergent. For this series, the absolute series is \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \), which diverges as it resembles the harmonic series.

Key concepts

Series Convergence

When dealing with infinite series, it's important to determine whether the series converges or diverges. Series convergence is the concept where the sum of the infinitely many terms of a series approaches a specific finite value. For the series to converge, the conditions of the convergence tests applied, like the Alternating Series Test, must be met. In our example, the given series is \[\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\]This is an alternating series because the terms alternate between positive and negative due to the \((-1)^n\) factor. To test for convergence, we applied the Alternating Series Test, ensuring that each term's absolute value decreases and approaches zero. Because the series meets these conditions, it converges.

Absolute Convergence

Absolute convergence delves a step deeper than regular convergence. A series is absolutely convergent if the series formed by taking the absolute values of its terms is convergent as well. This means we consider the series:\[\sum_{n=1}^{\infty} \left| \frac{(-1)^n}{\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\]In the example, this new series represents the harmonic series, which diverges. Since the absolute series diverges, the original series is only conditionally convergent, not absolutely. Absolute convergence is important because it implies convergence regardless of the order of terms. This property does not hold for conditionally convergent series.

Divergent Series

A divergent series is one where the sum of its terms grows indefinitely or doesn't settle to a singular value. For example, the harmonic series:\[\sum_{n=1}^{\infty} \frac{1}{n}\]is a well-known example of a divergent series, growing without bounds as more terms are added. In our original exercise, when evaluating absolute convergence, we ended up with a divergent series because:\[\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\]resembles the form of the harmonic series. Divergence is a critical concept because it reveals that the infinite sum doesn't converge to a finite number and can signal different behaviors within a sequence or series, guiding further analysis for handling such series.