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The harmonic series is one of the most fundamental divergent series that appears often in mathematics. This series is expressed as \( \sum \frac{1}{n} \), where \( n \) is a positive integer starting from 1 and incrementing by 1 for each following term. Each term is the reciprocal of a natural number. While the terms become very small as \( n \) increases, the harmonic series grows extraordinarily slowly yet without bound. This characteristic leads to its divergence. In our example, the new series formed was \( \sum \frac{1}{n} \), exactly the harmonic series. Despite the sum \( \sum \frac{1}{n} \) appearing to approach a finite limit due to the diminishing size of each term, it continues to grow larger indefinitely as more terms are added. This subtle property is a classic example of a series that diverges.

A convergent series is one where the sum of all its terms approaches a specific finite value as more terms are added. The series \( a_n = \sum \frac{1}{n^2} \) exemplifies a convergent series, demonstrating a situation where the infinite sum of diminishing terms results in a finite number.In mathematics, for a series \( \sum a_n \) to be convergent, the limit of the partial sums of the series must exist and be finite. For example, the series \( \sum \frac{1}{n^2} \) converges to \( \frac{\pi^2}{6} \). This means as you add more terms of \( \frac{1}{n^2} \), the total sum gets closer and closer to \( \frac{\pi^2}{6} \), never exceeding this value and never failing to approach it. The convergence of a series can be determined through several tests such as the comparison test, ratio test, and integral test, which help in identifying the behavior of the series and whether it will converge or diverge.

On the flip side, a divergent series is one that does not converge. In other words, the series does not approach a finite number as more terms are added. Instead, it either grows indefinitely or does not settle at any specific value.The series \( \sum \frac{1}{n} \) known as the harmonic series is a well-known divergent series. Even though its terms \( \frac{1}{n} \) keep getting smaller and smaller, the sum continues to increase forever. When talking about divergences, it is crucial to realize that a divergent series has no limit. This means attempts to assign it a fixed value would not reflect the true behavior of the series. In mathematical terms, no matter how many terms you add up in the series, it will never approach a specific number, showing that it is divergent and its sum is ultimately infinite.