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A geometric series is a series of the form \(a + ar + ar^2 + ar^3 + \ldots\), where \(a\) is the first term, and \(r\) is the common ratio between successive terms. The nature of the common ratio \(r\) greatly influences whether the series converges or diverges.

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

It is important to determine the common ratio as it helps us find the sum of the series when it converges. The sum of an infinite geometric series is given by the formula:\[S = \frac{a}{1-r}\]For example, for the series \(7 \sum_{k=0}^{\infty} \left(\frac{-1}{6}\right)^{k}\), the first term \(a\) is 7 and the common ratio \(r\) is \(-\frac{1}{6}\). Since \(|r| = \frac{1}{6} < 1\), the series converges and its sum can be calculated. Applying the formula correctly helps simplify complex series calculations.

An alternating series is one where the signs of the terms alternate, typically represented as \((-1)^n a_n\) or \((-1)^{n-1} a_n\). This alternation in sign is key in determining convergence through the Alternating Series Test.Key properties of an alternating series:

• The terms of the series must decrease in absolute value as the series progresses.
• As \(n\) approaches infinity, the absolute value of the terms must approach zero, i.e., \(\lim_{n \to \infty} a_n = 0\).

When these conditions are met, the series converges. For the series \(\sum_{k=1}^{\infty} (-1)^{k-1} \frac{7}{6^{k-1}}\), we see it alternates signs and decreases in magnitude, fulfilling these criteria, thus ensuring convergence. Understanding these conditions is crucial for analyzing the behavior of alternating series.

The sum of an infinite series can be a complex topic, especially depending on the type of series. For geometric series and alternating series, distinct formulas and tests help find these sums when conditions for convergence are met.- **Geometric Series Sum**: If the series is geometric and converges, the sum is given by \(S = \frac{a}{1-r}\). This formula simplifies finding sums of convergent geometric series efficiently.- **Alternating Series**: While there's no direct formula like geometric series, convergence can be established using the Alternating Series Test. Once convergence is confirmed, the remainder/error after \(n\) terms is less than the first omitted term, providing a means to approximate the sum.For the specific case of \(7 \sum_{k=0}^{\infty} \left(\frac{-1}{6}\right)^{k}\), we use the geometric series formula once convergence is confirmed. The sum simplifies to 6, illustrating that with correct application of these principles, analyzing infinite series becomes more approachable.