Problem 17 - Calculus Single Variable

Question

In each of Exercises \(17-28,\) use the Comparison Test for Divergence to show that the given series diverges. State the series that you use for comparison and the reason for its divergence. $$ \sum_{n=1}^{\infty} \frac{1}{2 n-1} $$

Step-by-Step Solution

Answer

The series diverges by comparison with the harmonic series, which is a known divergent series.

Step 1: Identify the given series

We are given the series \( \sum_{n=1}^{\infty} \frac{1}{2n-1} \). We need to analyze this series for divergence using the Comparison Test for Divergence.

Step 2: Choose a series for comparison

Let's compare it with the well-known harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is known to diverge.

Step 3: Write the comparison inequality

Observe that for all \( n \geq 1 \), \( 2n - 1 < n \) implies \( \frac{1}{2n-1} > \frac{1}{n} \). This shows that each term of the given series is greater than the corresponding term of the harmonic series when \( n \geq 1 \).

Step 4: Apply the Divergence Test

Since the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges and \( \frac{1}{2n-1} > \frac{1}{n} \) for each term, by the Comparison Test for Divergence, the series \( \sum_{n=1}^{\infty} \frac{1}{2n-1} \) also diverges.

Step 5: Conclusion

We have shown that the given series \( \sum_{n=1}^{\infty} \frac{1}{2n-1} \) diverges by comparing it to the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is divergent. Hence, the series diverges.

Key Concepts

Comparison Test

When analyzing whether a series diverges or converges, the **Comparison Test** can be a very handy tool. This test requires comparing your series, called \( a_n \), with a second series, \( b_n \), where the behavior—whether it converges or diverges—is already known. Whenever \( 0 \leq a_n \leq b_n \) and the series \( \sum b_n \) converges, then \( \sum a_n \) must also converge. The opposite is true for divergence: If \( a_n \geq b_n \) and \( \sum b_n \) diverges, then \( \sum a_n \) will diverge too. This is particularly useful because it allows us to confidently say whether our series behaves in the same way as our comparison series. To utilize the Comparison Test effectively, ensure your choice of comparison series \( b_n \) is logical and shares a similar form to \( a_n \). In practice, this often involves using well-known series such as geometric or p-series.

Harmonic Series

The **Harmonic Series** is one of the most famous divergent series, written as \( \sum_{n=1}^{\infty} \frac{1}{n} \). Despite each term becoming smaller as \( n \) increases, the series does not summate to a finite limit. As you continue to add these ever-smaller fractions, the total never stabilizes to a specific value; it grows unbounded. This outcome may seem counterintuitive because the terms \( \frac{1}{n} \) progress towards zero. The divergence of the harmonic series makes it an excellent benchmark for the Comparison Test's divergence purposes.To find out why this series diverges, consider grouping the terms:- The first term alone equals 1. - The next two \,terms \( \left( \frac{1}{2} + \frac{1}{3} \right) \) surpass \( \frac{1}{2} \). - The next four terms \( \left( \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} \right) \) are greater than \( \frac{1}{2} \). This process shows a pattern of exceeding any fixed bound upon summation, confirming its divergence.

Divergence Test

The **Divergence Test** is a straightforward method to analyze series. It's based on a simple rule—if the terms of a series do not approach zero as \( n \) approaches infinity, then the series \( \sum a_n \) will not converge. However, it's crucial to note that this test can only tell you if a series diverges, not if it converges.To apply the test, take the limit of \( a_n \). If \( \lim_{{n \to \infty}} a_n eq 0 \), the series diverges. However, if the limit equals zero, the test remains inconclusive, and further analysis will be required using other methods such as the Comparison Test.When combined with other tests like the Comparison Test as seen in our original exercise, the Divergence Test acts as a basic initial check, ensuring our base conditions for divergence checking are satisfied before proceeding with more sophisticated examination.