Question

. \(\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{\sqrt{n^{3}+1}}\)

Short answer

The series converges.

Step-by-step solution

Identify the Series

Recognize the given series as \(\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{\sqrt{n^{3}+1}}\)

Test for Absolute Convergence

Examine the absolute value of the general term \(a_{n} = \left| \frac{(-1)^{n} n}{\sqrt{n^{3}+1}} \right| = \frac{n}{\sqrt{n^{3}+1}}\). Note that \(a_{n}\) does not tend to zero as \(n\) approaches infinity because \(\frac{n}{\sqrt{n^{3}+1}} \approx \frac{n}{n^{3/2}} = \frac{1}{\sqrt{n}}\), which approaches zero.

Apply Alternating Series Test

Since the series is alternating and \(\left| \frac{(-1)^{n} n}{\sqrt{n^{3}+1}} \right| \to 0\) as \(n \to \infty,\) check the additional condition: \(a_{n+1} \leq \a_{n}\). For large \(n,\) \(\left( \frac{(n+1)}{\sqrt{(n+1)^{3}+1}} \right) \leq \left( \frac{n}{\sqrt{n^{3}+1}} \right)\). This indicates the series is decreasing and confirms that it converges by the Alternating Series Test.

Conclusion

Conclude that the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{\sqrt{n^{3}+1}}\) converges by the Alternating Series Test since both conditions are satisfied.

Key concepts

headline of the respective core concept

The convergence of a series is an essential concept in calculus and mathematical analysis. When we talk about the convergence of a series, we are determining whether the sum of the infinite terms results in a finite value. In simpler terms, we check if we keep adding the terms of the series, will we ever approach a fixed number or not? If we do, we say that the series converges; otherwise, the series diverges.
For instance, in the given problem, we deal with the series \(\frac{(-1)^n n}{\text{sqrt}{n^3}+1}\). Determining its convergence involves checking the behavior of the terms as \(n\) becomes very large. Various tests, such as the Alternating Series Test, which was applied in this solution, help us evaluate the convergence properties.

headline of the respective core concept

Absolute convergence is a stronger form of convergence for a series. If a series \(\text{sum a}_n\) is said to be absolutely convergent, this means that the series \(\text{sum |a}_n|\) (the series of the absolute values of its terms) also converges.
Absolute convergence can imply regular convergence, but the reverse isn't true. For the given series \(\text{sum (-1)^n n / sqrt(n^3) + 1}\), we checked for absolute convergence. However, as shown in the step-by-step solution, the absolute values of terms still approach zero, but more gradual, affirming the series isn’t absolutely convergent.
Remember: Absolute convergence is crucial because it guarantees convergence irrespective of the arrangement of terms, which is crucial for series manipulations.

headline of the respective core concept

Series comparison tests are another powerful tool for evaluating the convergence of a series. They involve comparing the series in question with a series whose convergence properties are already known. This method is based on relative behavior.
There are two main types:

• Limit Comparison Test: If the limit of the ratio of the terms of two series is finite and positive, both series will either converge or diverge.
• Direct Comparison Test: If every term of one series is less than the corresponding term of another, and if the larger series converges, then so must the smaller.

In the given exercise, the comparison to simpler forms like \(n/ n^3/2\) helped in realizing that our series terms \(n/sqrt(n^3)\), which tend to zero supports convergence.
This method is useful when dealing with complex series by breaking them down into more manageable pieces.

headline of the respective core concept

Mathematical proofs are essential in establishing the truth behind mathematical statements rigorously. They involve a series of logical steps which derive a conclusion from initial assumptions or known truths.
In the context of series convergence, certain proofs like the Alternating Series Test require verifying specific conditions:

• The terms must alternate in sign.
• The absolute value of terms must decrease monotonically.
• The limit of the terms must approach zero.

For the given series \(\text{sum (-1)^n n / sqrt(n^3)+1}\), these conditions were thoroughly checked. As a result, the proof steps confirmed the series' convergence.
Mastering different mathematical proofs helps in understanding and forming insights into why a series behaves or adheres to certain properties. This foundation is critical for advanced studies in mathematics and related fields.