Question

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} n\left(\tan ^{-1}(n+1)-\tan ^{-1} n\right)\) (Hint: Use Mean Value Theorem.)

Short answer

The series neither converges absolutely nor conditionally.

Step-by-step solution

Identify the series

The given series is \(\sum_{n=1}^{\infty}(-1)^{n+1} n\left(\tan^{-1}(n+1)-\tan^{-1} n\right)\). This series is alternating because of the \((-1)^{n+1}\) term.

Apply the Mean Value Theorem

For the function \(f(x) = \tan^{-1}(x)\), the derivative is \(f'(x) = \frac{1}{1+x^2}\). According to the Mean Value Theorem, \(\tan^{-1}(n+1) - \tan^{-1}(n) = f'(c_n)(n+1-n)\) for some \(c_n \in (n, n+1)\). Hence, \(\tan^{-1}(n+1) - \tan^{-1}(n) = \frac{1}{1+c_n^2}\).

Simplify the series terms

Substitute the result from Step 2 into the series terms: \(n\left(\tan^{-1}(n+1) - \tan^{-1}(n)\right) = \frac{n}{1+c_n^2}\). Now, the series is \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{1+c_n^2}\).

Evaluate absolute convergence

To check for absolute convergence, consider the series \(\sum_{n=1}^{\infty} \left|(-1)^{n+1} \frac{n}{1+c_n^2}\right| = \sum_{n=1}^{\infty} \frac{n}{1+c_n^2}\). For \(n \to \infty\), \(\frac{n}{1+c_n^2} \sim n\), diverging as it grows without bound. So, the series does not converge absolutely.

Evaluate conditional convergence

To check for conditional convergence, we need to analyze whether the series \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{1+c_n^2}\) meets the criteria of the Alternating Series Test: (i) \(b_n = \frac{n}{1+c_n^2}\) decreases, and (ii) \(\lim_{n \to \infty} b_n = \lim_{n \to \infty} \frac{n}{1+c_n^2} = \infty\). Since criterion (ii) fails, the series does not converge conditionally.

Key concepts

Alternating Series

An alternating series is a series where the terms alternate in sign. This means that every other term is negative, creating a sequence that looks like positive, negative, positive, negative, and so on. A classic example of alternating series can be written as \(\sum_{n=1}^{\infty} (-1)^{n+1} a_n\),where \((-1)^{n+1}\) determines the sign of each term, flipping it between positive and negative. The alternating nature can often help series converge, even if the terms don't become zero very quickly.

When analyzing convergence, it's common to use the Alternating Series Test. This test provides two conditions that if satisfied, show that the series converges:

• The absolute values of the terms \(b_n\) decrease steadily, or decrease 'monotonically.'
• The limit of \(b_n\) as \(n\) approaches infinity is zero.

For our exercise, we looked at whether the alternating series converged by evaluating these conditions, which we found it did not.

Mean Value Theorem

The Mean Value Theorem (MVT) is a crucial concept in calculus used to analyze functions and their behavior over an interval. The theorem states that for any given section of a smooth, continuous function, there is at least one point where the function's slope (derivative) equals the average rate of change over that interval.

Mathematically, if \(f(x)\) is continuous over the interval \([a, b]\) and differentiable over \((a, b)\), then there exists some \(c\) in \((a, b)\) such that:\[f'(c) = \frac{f(b) - f(a)}{b-a}\]In our exercise, the Mean Value Theorem helped simplify the expression \(\tan^{-1}(n+1) - \tan^{-1}(n)\). By applying the theorem, we expressed this difference using the derivative of \(\tan^{-1}(x)\). This allowed us to further analyze the series for convergence.

Absolute Convergence

Absolute convergence is a straightforward concept in series analysis. A series is said to converge absolutely if the series formed by taking the absolute values of its terms also converges. If a series \(\sum_{n=1}^{\infty} a_n\) converges absolutely, then the series \(\sum_{n=1}^{\infty} |a_n|\) also converges.

Checking for absolute convergence often involves assessing the series without considering the signs of its terms. In our exercise, evaluating for absolute convergence involved considering the positive series \(\sum_{n=1}^{\infty} \frac{n}{1+c_n^2}\). However, since this does not converge, the original series doesn’t converge absolutely either. Absolute convergence is a strong form of convergence that guarantees regular convergence, but it wasn't applicable here.

Conditional Convergence

Conditional convergence occurs when a series converges, but does not converge absolutely. This can happen particularly with alternating series, where the changing signs help the overall series to stabilize and converge.

In the context of our exercise, we attempted to see if the series converged conditionally after finding it didn't converge absolutely. By deploying the Alternating Series Test, we intended to test if the terms decreased to zero. Generally, a series can be conditionally convergent if it fails absolute convergence but passes the alternating series conditions. However, our specific series didn't meet these prerequisites for conditional convergence, indicating that it neither converged absolutely nor conditionally.