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

Short answer

The series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n!}}\) converges absolutely.

Step-by-step solution

Identify the Alternating Series

The given series is \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n!}}\). This is an alternating series because the terms include \((-1)^{n+1}\), which causes the series to alternate signs.

Apply the Alternating Series Test

For the alternating series \(\sum (-1)^n a_n\) to converge, \(a_n\) must be positive, decreasing, and its limit as \(n\) approaches infinity must be zero. Here, \(a_n = \frac{1}{\sqrt{n!}}\) is positive. We check if \(a_n\) is decreasing and if \(\lim_{n \to \infty} a_n = 0\).

Check for Decreasing Behavior

Determine if \(a_n = \frac{1}{\sqrt{n!}}\) is decreasing. Since \(n!\) grows very fast, \(\sqrt{n!}\) increases, thus \(\frac{1}{\sqrt{n!}}\) decreases as \(n\) increases.

Take the Limit of aₙ

Calculate \(\lim_{n \to \infty} \frac{1}{\sqrt{n!}}\). As \(n!\) tends to infinity with growing \(n\), \(\frac{1}{\sqrt{n!}}\) becomes infinitely small. Therefore, \(\lim_{n \to \infty} \frac{1}{\sqrt{n!}} = 0\).

Conclusion of Alternating Series Test

Since \(a_n\) is positive, decreasing, and \(\lim_{n \to \infty} a_n = 0\), the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n!}}\) converges.

Determine Absolute Convergence

To determine if the convergence is absolute, check the series \( \sum_{n=1}^{\infty} \left| \frac{(-1)^{n+1}}{\sqrt{n!}} \right| = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}} \).

Analyze \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}}\)

This series must be tested for convergence. Due to the rapid growth of \(n!\), the terms \(\frac{1}{\sqrt{n!}}\) decrease very quickly, leading to the conclusion that the series converges absolutely.

Determine Type of Convergence

Since the series converges both originally and absolutely, the convergence of \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n!}}\) is absolute.

Key concepts

Convergence

In mathematics, convergence is a crucial concept when analyzing series and sequences. A series is a sum of terms that follows a specific pattern or rule.
Convergence happens when the sum of these terms approaches a finite number as we keep adding more terms.
For the series to converge, the terms must continuously get smaller and eventually approach zero. Understanding convergence helps decide if a series will settle at a certain value or explode to infinity. A convergent series aligns closer to a definite number as more terms are added.

Alternating Series Test

The Alternating Series Test, a tool used to assess the convergence of a series, plays a key role with alternating series. These are series where the sign of each term switches between positive and negative.For a series \[\sum (-1)^n a_n\]to pass the Alternating Series Test and thus converge, the following criteria must be met:

• Each term, denoted as \(a_n\), should be positive.
• The terms must be steadily decreasing in absolute value.
• The limit of \(a_n\) as \(n\) approaches infinity should be zero, meaning the terms get infinitesimally small.

When these conditions are verified, you can confidently say the alternating series converges.

Absolute Convergence

Absolute convergence of a series means that the series converges even when we take the absolute value of each term.Consider the series \(\sum_{n=1}^{\infty} a_n\). It converges absolutely if the series of absolute values, \(\sum_{n=1}^{\infty} |a_n|\), also converges.
This condition ensures that the series behaves well, regardless of the arrangement of its terms.To determine absolute convergence:

• Transform each term of the series to its absolute value.
• Test the new series with convergence criteria, such as the Alternating Series Test.

If both the original and absolute series converge, the convergence is absolute, which indicates a stronger form of convergence compared to conditional convergence.

Factorial Growth

Factorial growth is a concept that refers to the rapid increase of the function \(n!\) as \(n\) increases. The symbol \(n!\) ("n factorial") means \(n \times (n-1) \times (n-2) \times \cdots \times 1\).Factorials grow very quickly, which significantly affects calculations, especially in series.For instance, in the series \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}}\), the growth of \(n!\) causes the terms \(\frac{1}{\sqrt{n!}}\) to become increasingly smaller for larger \(n\).This fast decay in the size of the terms is a factor in proving the convergence of the series using tests like the Alternating Series Test and criteria for absolute convergence.