Problem 32 - Single Variable Calculus

Question

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{2^{n}}{3 \cdot 5 \cdot 7 \cdots(2 n+1)}\)

Step-by-Step Solution

Answer

The given series \(\sum_{n=1}^{\infty}(-1)^{n} \frac{2^{n}}{3 \cdot 5 \cdot 7 \cdots(2 n+1)}\) is absolutely convergent, based on the Ratio Test.

Step 1: Apply the Ratio Test for Absolute Convergence

To apply the Ratio Test, we need to compute the limit of the ratio of consecutive absolute terms: \[\lim_{n\to\infty}\frac{\left|\frac{(-1)^{n+1} 2^{n+1}}{3\cdot5\cdot7\cdots(2(n+1)+1)}\right|} {\left|\frac{(-1)^n 2^n}{3\cdot5\cdot7\cdots(2n+1)}\right|} \] First, we simplify the expression inside the limit: \[\frac{2^{n+1}}{3\cdot5\cdot7\cdots(2(n+1)+1)} \cdot \frac{3\cdot5\cdot7\cdots(2n+1)}{2^n} = \frac{2}{2n+3}\] Now, evaluate the limit as n approaches infinity: \[\lim_{n\to\infty}\frac{2}{2n+3} = 0\] Since this limit is less than 1, the series is absolutely convergent.

Step 2: Determine the Type of Convergence

Since the series is absolutely convergent, we know that it is convergent. Therefore, it is not necessary to test for conditional convergence or divergence.

Step 3: Conclusion

The given series \(\sum_{n=1}^{\infty}(-1)^{n} \frac{2^{n}}{3 \cdot 5 \cdot 7 \cdots(2 n+1)}\) is absolutely convergent.

Key Concepts

Ratio Test

The Ratio Test is a helpful tool for determining the convergence of a series. It's especially useful for series with terms involving powers or factorials. To apply the Ratio Test, we consider the absolute value of the ratio between consecutive terms of the series. We denote this by the formula:

\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]

• If \(L < 1\), the series converges absolutely.
• If \(L > 1\), the series is divergent.
• If \(L = 1\), the test is inconclusive.

In the exercise, the Ratio Test reveals that the limit \(L = 0\), which means the series converges absolutely. This test simplifies the process, giving a clear indication about the convergence without heavy computation.

Absolutely Convergent

A series is said to be absolutely convergent if the series of the absolute values of its terms converges. In simpler terms, if you take each term of the series, ignore its sign, and the resulting series sums to a finite number, then the original series is absolutely convergent.

Why is this important?

• Absolute convergence implies convergence. If a series is absolutely convergent, it is also convergent in the standard sense.
• This property is extremely powerful, as it guarantees the behavior of the series regardless of the arrangement of terms.
• In the original exercise, using the Ratio Test, we determined that our series converges absolutely because the limit of the ratio test was less than 1.

Absolutely convergent series are the most stable form of convergent series, making them highly important in analysis.

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. You can think of it as adding up endless numbers, one after another. Mathematically, it's represented as:

\[\sum_{n=1}^{\infty} a_n\]
Where each \(a_n\) represents a term in the series. The key question for any infinite series is whether it converges or diverges:

• Convergent: If adding all its terms will approach a specific value.
• Divergent: If the sum does not approach any unique value.

In our problem, the series was infinite because it stretched from \(n=1\) to infinity. We discovered it was convergent through its absolute convergence. Therefore, the infinite nature of the series doesn't affect its ability to converge; only the nature of its terms does.

Conditional Convergence

Conditional convergence is a fascinating concept in the study of series. It occurs when a series is convergent, but not absolutely convergent.

In other words, if a series converges only when the terms are allowed to keep their sign, but not when converted to their absolute values, it is conditionally convergent.

• The most common example of a conditionally convergent series is the alternating harmonic series.
• This happens because there is a delicate balance between the signs and the magnitudes of terms that allows the series to approach a limit but would otherwise not stabilize.

The given exercise aimed to identify whether the series was absolutely or conditionally convergent. Since the Ratio Test resulted in absolute convergence, we did not need to check for conditional convergence as it's already a stronger result.