Question

An alternating series \(\sum_{n=i}^{\infty} a_{n}\) is given. (a) Determine if the series converges or diverges. (b) Determine if \(\sum_{n=0}^{\infty}\left|a_{n}\right|\) converges or diverges. (c) If \(\sum_{n=0}^{\infty} a_{n}\) converges, determine if the convergence is conditional or absolute. $$\sum_{n=2}^{\infty}(-1)^{n} \frac{n}{\ln n}$$

Short answer

The given series diverges; hence, it is neither conditionally nor absolutely convergent.

Step-by-step solution

Check Alternating Series Convergence

To determine if the given series \(\sum_{n=2}^{\infty}(-1)^{n} \frac{n}{\ln n}\) converges, we use the Alternating Series Test. This test states that a series \(\sum (-1)^n a_n\) converges if \(a_n\) is positive, decreasing, and \(\lim_{n \to \infty} a_n = 0\).For this series, \(a_n = \frac{n}{\ln n}\). While \(\lim_{n \to \infty} \frac{n}{\ln n} = \infty\), which does not satisfy the condition of the Alternating Series Test that the terms approach zero as \(n\) approaches infinity. Therefore, the series diverges by the Alternating Series Test.

Examine Absolute Convergence

Next, consider the series \(\sum_{n=2}^{\infty} \left| (-1)^n \frac{n}{\ln n} \right| = \sum_{n=2}^{\infty} \frac{n}{\ln n}\). We examine whether this series converges by the Integral Test. Consider the function \(f(x) = \frac{x}{\ln x}\) which is positive for \(x > 1\). We check the integral \(\int_{2}^{\infty} \frac{x}{\ln x} \, dx\). As \(\ln x\) grows slower than \(x\), the function \(\frac{x}{\ln x}\) does not decrease to 0, and hence, \(\int_{2}^{\infty} \frac{x}{\ln x} \, dx\) diverges. Thus, \(\sum_{n=2}^{\infty} \frac{n}{\ln n}\) diverges.

Determine if any Convergence is Conditional or Absolute

The series \(\sum_{n=2}^{\infty}(-1)^{n} \frac{n}{\ln n}\) does not converge, so it is neither conditionally nor absolutely convergent. Conditional convergence requires that the series is convergent but the series of absolute values diverges, which is not the case here since the original series itself diverges.

Key concepts

Alternating Series Test

The Alternating Series Test is a useful tool in determining the convergence of a specific form of series known as alternating series. These are series where the terms alternate in sign, such as \(-1^{n} \rac{n}{\ln n}\). To conclude convergence using this test, you should check for three key conditions:

- The terms \(a_n\) must be positive.
- The sequence of terms must be decreasing, which means \(a_{n+1} \leq a_{n}\) for all \(n\).
- The limit of the terms \(a_n\) as \(n\) approaches infinity should be zero, i.e., \(\lim_{n \to \infty} a_n = 0\).

In our solution, although the series has alternating signs, the term \(a_n = \frac{n}{\ln n}\) fails one of these conditions. Specifically, \(\lim_{n \to \infty} \frac{n}{\ln n}\) does not equal zero and instead approaches infinity. Hence, using the Alternating Series Test, we determine that the series diverges.

Absolute Convergence

Absolute convergence refers to a series \(\sum a_n\) in which the series of its absolute values, \(\sum |a_n|\), is convergent. To determine this, it is often helpful to use tests like the Integral Test, Comparison Test, or known convergent series for comparison.

For our series \(\sum (-1)^{n} \frac{n}{\ln n}\), checking absolute convergence involves examining \(\sum \frac{n}{\ln n}\), which is simply the series of absolute values without signs. By attempting the Integral Test on \(f(x) = \frac{x}{\ln x}\), we found that the integral \(\int_{2}^{\infty} \frac{x}{\ln x} \, dx\) diverges. This behavior implies that the series \(\sum \frac{n}{\ln n}\) also diverges, confirming that the original alternating series does not converge absolutely. Absolute convergence would have required this series to be convergent, but since it isn't, the series cannot be considered absolutely convergent.

Integral Test

The Integral Test is a method for determining the convergence or divergence of infinite series. This test relates a series to an improper integral. If you find the convergence of \(\int_{a}^{\infty} f(x) \, dx\), it can conclude whether the series \(\sum_{n=a}^{\infty} a_n\) converges as well, provided \(f(x)\) is positive, continuous, and decreasing.

In assessing the series \(\sum \frac{n}{\ln n}\), our function is \(f(x) = \frac{x}{\ln x}\). This function does meet the requirements of the Integral Test. However, since \(\ln x\) rises slower than \(x\), the digestion of the integral \(\int_{2}^{\infty} \frac{x}{\ln x} \, dx\) establishes divergence, indicating that the series \(\sum \frac{n}{\ln n}\) diverges too.

The convergence of the integral directly informs us about the divergence of the related series, serving as a handy check in cases like these. This is crucial information that directly affects the understanding of the series' convergence properties.