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A convergent series is one where the sum of its terms approaches a specific finite number, known as the limit, as more and more terms are added. Convergence happens when the partial sums of the series become closer and closer to a certain value. This means that as you take the sum of a very large number of terms, the result gets very close to a certain number rather than infinitely increasing or decreasing. For example, the series \(\sum \frac{1}{2^n}\) is convergent because each additional term \(\frac{1}{2^n}\) gets smaller and smaller, causing the overall sum to approach a fixed value.
Some key points about convergent series:

• They have a finite sum.
• Adding more terms brings the sum closer to a certain number.
• Geometric series with a ratio \(r < 1\) are examples of convergent series.

Understanding the concept of convergence is essential in many areas of mathematics as it tells us about the behavior of the series in the long term.

A geometric series is a series with a constant ratio between consecutive terms. The general form of a geometric series is \(a + ar + ar^2 + ar^3 + \ldots\), where \(a\) is the first term and \(r\) is the common ratio.
Geometric series can either converge or diverge, depending on the value of the common ratio \(r\):

• If \(|r| < 1\), the series converges and the sum can be calculated using the formula \(S = \frac{a}{1-r}\).
• If \(|r| \geq 1\), the series diverges, meaning the sum is not finite.

For example, the series \(\sum \frac{1}{2^n}\) is a geometric series with a ratio \(r = \frac{1}{2}\), which is less than 1, thus it converges. On the other hand, \(\sum \left(\frac{3}{2}\right)^n\) diverges because its ratio \(r = \frac{3}{2}\) is greater than 1. Recognizing whether a series is geometric helps in quickly determining converge or divergence.

A series is divergent if its terms do not approach any finite sum as more and more terms are added. Instead of getting closer to a specific value, the sum grows indefinitely or oscillates without settling into a finite limit. This happens when the partial sums of a series keep increasing or never stabilize to one value when added infinitely.
Key signs of divergence include:

• The partial sums increase without bound.
• The terms of the series do not get progressively smaller.
• For geometric series, if the ratio \(r\) is \(|r| \geq 1\), the series diverges.

The exercise example \(\sum \left(\frac{3}{2}\right)^n\) diverges because each term becomes larger and the sum increases without limit. It illustrates that even if each series in a fraction converges, the fraction itself may not.

The ratio test is a handy tool to determine whether a series is convergent or divergent. This test involves calculating the limit of the absolute value of the ratio of a term and the next term in the series. Suppose \(a_n\) is a term in the series; we consider the series \(\sum a_n\):

• Compute \(L = \lim_{n \to \infty}\left| \frac{a_{n+1}}{a_n} \right|\).
• If \(L < 1\), the series \(\sum a_n\) converges.
• If \(L > 1\), the series \(\sum a_n\) diverges.
• If \(L = 1\), the test is inconclusive, meaning other methods are required.

The ratio test is particularly useful for series where terms involve exponential or factorial functions. In the original solution, the divergence of \(\sum \left(\frac{3}{2}\right)^n\) could be quickly determined using the ratio test, as the common ratio \(\frac{3}{2}\) results in \(L > 1\). Thus, understanding and applying the ratio test can simplify the task of assessing series behavior.