Problem 2 - Calculus Early Transcendentals

Question

Is it true that if the terms of a series of positive terms decrease to zero, then the series converges? Explain using an example.

Step-by-Step Solution

Answer

Answer: No, it is not always true. Although the terms of both the given examples decrease to zero, one series converges while the other diverges. It is essential to analyze the convergence or divergence using appropriate methods, such as the sum formula or integral test, to determine if a series converges or not.

Step 1: Understanding Convergence and Divergence

When we talk about a series converging or diverging, we are essentially referring to the sum of its terms. A series converges if the sum of its terms approaches a finite value, and it diverges if the sum of its terms goes to infinity or doesn't have a finite limit.

Step 2: Example 1 - Converging Series

Let's consider a series of positive terms that decrease to zero: The geometric series \[\sum_{n=1}^{\infty} \frac{1}{2^n}\] To check whether this series converges or not, we calculate the sum of the infinite geometric series using the formula \[S = \frac{a_1}{1 - r}\] where \(a_1\) is the first term, and \(r\) is the common ratio. In this case, \(a_1 = \frac{1}{2}\) and \(r = \frac{1}{2}\). Therefore, \[S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1\] Since the sum is finite, this series converges.

Step 3: Example 2 - Diverging Series

Let's consider another series of positive terms that decrease to zero: The harmonic series \[\sum_{n=1}^{\infty} \frac{1}{n}\] To check whether this series converges or not, we use the integral test, which states that the series converges if and only if the corresponding improper integral converges: \[\int_{1}^{\infty} \frac{1}{x} dx\] Calculate the integral: \[\int_{1}^{\infty} \frac{1}{x} dx = \lim_{b \to \infty} (\ln(x) \Big|_1^b)\] \[\lim_{b \to \infty} (\ln(b) - \ln(1)) = \lim_{b \to \infty} \ln(b) = \infty\] Since the improper integral diverges to infinity, the harmonic series also diverges.

Step 4: Conclusion

Although the terms of both examples decrease to zero, one series converges while the other diverges. Therefore, it's not always true that if the terms of a series of positive terms decrease to zero, the series converges.