In the study of sequences and series, it's important to understand the concept of convergence and divergence. A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases. On the other hand, a series diverges if the sum of its terms does not approach a finite value and instead either grows infinitely or oscillates.

There are several tests that can help determine whether a series converges or diverges. One commonly used test is the comparison test, which compares the given series to a known convergent or divergent series. Another test is the ratio test, which examines the ratio of consecutive terms within the series. If the ratio is less than one, the series converges, while a ratio greater than one indicates divergence. Additionally, the integral test can be employed to determine convergence by comparing the series to the integral of a related function.

Some special types of series that require specific tests for convergence include harmonic series and alternating series. The harmonic series diverges, as its terms do not approach a finite value. Alternating series, however, have alternating signs for their terms and often converge if they satisfy certain conditions, such as decreasing magnitude.

Remember, understanding convergence and divergence of series is essential for analyzing the behavior of mathematical models and calculations in various fields. So keep practicing and exploring more examples to strengthen your grasp on this topic!