Question

State whether the given series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{2 n-1}$$

Short answer

The series diverges.

Step-by-step solution

Identify the Form of the Series

Recognize the series as \( \sum_{n=1}^{\infty} \frac{1}{2n-1} \), which has the form \( \sum_{n=1}^{\infty} \frac{1}{a_n} \) with \( a_n = 2n-1 \). This is similar to the harmonic series when \( a_n = n \).

Compare with the Harmonic Series

The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is known to diverge. In our series, \( 2n-1 \sim n \) for large \( n \), which implies that the terms \( \frac{1}{2n-1} \) behave similarly to \( \frac{1}{n} \).

Use the Limit Comparison Test

Compute the limit \( \lim_{n \to \infty} \frac{\frac{1}{2n-1}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{2n-1} = \lim_{n \to \infty} \frac{1}{2 - \frac{1}{n}} = \frac{1}{2} \). Since this limit is finite and positive, the Limit Comparison Test tells us that \( \sum_{n=1}^{\infty} \frac{1}{2n-1} \) converges or diverges in the same manner as the harmonic series.

Conclusion Based on the Convergence Test

Since the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges and we found a positive finite limit in the Limit Comparison Test, \( \sum_{n=1}^{\infty} \frac{1}{2n-1} \) also diverges.

Key concepts

Harmonic Series

The harmonic series is one of those fundamental concepts in mathematics that frequently appears in discussions about convergence and divergence of infinite series. It is represented as: \[ \sum_{n=1}^{\infty} \frac{1}{n} \]This series is called 'harmonic' because its terms are similar to the harmonic mean of consecutive numbers. A fascinating aspect of the harmonic series is that it diverges, meaning that even though its terms get smaller and smaller, the sum grows without bound as more terms are added. Understanding this behavior is crucial when comparing it with other series to determine their convergence or divergence behavior.The divergence of the harmonic series is a key point of reference because any series that behaves similarly in terms of the growth of its terms will likely exhibit the same infinite behavior. This insight allows us to use the concept of the harmonic series when applying comparison tests to other series.

Limit Comparison Test

The Limit Comparison Test is a powerful tool for determining the convergence or divergence of series. It compares the series in question to another series whose behavior is already known, such as the harmonic series. Here is how it works:

• Consider two series \( \sum a_n \) and \( \sum b_n \).
• Compute the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If \( L \) is finite and positive, both series converge or diverge together.

In our exercise, the series \( \sum_{n=1}^{\infty} \frac{1}{2n-1} \) was compared to the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \). By computing the limit:\[ L = \lim_{n \to \infty} \frac{\frac{1}{2n-1}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{2n-1} = \lim_{n \to \infty} \frac{1}{2 - \frac{1}{n}} = \frac{1}{2} \]Since \( L \) is 0.5, which is positive and finite, the Limit Comparison Test tells us that both the given series and the harmonic series diverge.

Divergence

Divergence in mathematical terms refers to the behavior of a series that continues to grow indefinitely as more terms are added. Such a series does not settle to a finite value but rather keeps expanding, which is precisely what happens with the harmonic series.In the context of our exercise, when we say the series \( \sum_{n=1}^{\infty} \frac{1}{2n-1} \) diverges, we're acknowledging that its sum becomes infinitely large as more terms are included. This conclusion comes from understanding that for this series, the terms do not decrease quickly enough to counterbalance the summation, ultimately leading to an unbounded growth.Divergence is an important concept because it informs us about the limitations of certain series in applications like physics and engineering, where having a sum that settles to a finite number is often critical. Recognizing when a series diverges helps mathematicians and scientists understand the behavior of systems modeled by such series.