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A geometric series is a series of terms in which each term is a constant multiple, known as the common ratio, of the previous term. To identify a geometric series, you can look for a sequence where each term is obtained by multiplying the preceding term by a fixed non-zero number, called the common ratio, denoted as \( r \). This gives the general form:

• \( a, ar, ar^2, ar^3, \, ... \)

The geometric series can be finite or infinite. For an infinite geometric series, it's important to know when it converges or diverges. An infinite geometric series converges if the absolute value of the common ratio is less than 1 (\( |r| < 1 \)), which results in a finite sum. Otherwise, the series diverges.

The formula for the sum \( S \) of an infinite geometric series is:

• \( S = \frac{a}{1 - r} \)

where \( a \) is the first term of the series. This formula allows us to efficiently find the sum of a convergent infinite geometric series.

An infinite series is a sum of infinitely many terms defined by a specific rule or pattern. It's denoted as \( \sum_{n=0}^{\infty} a_n \), where each \( a_n \) represents the terms of the series. When dealing with infinite series, a crucial aspect is determining whether a series converges or diverges. If a series converges, the sum of its terms approaches a specific finite value as the number of terms increases indefinitely. On the other hand, if a series diverges, the sum does not approach a finite value.

There are several tests used to determine convergence:

• Comparison Test: Compare with a known convergent or divergent series.
• Ratio Test: Useful for series where terms involve factorials or powers.
• Root Test: Similar to the ratio test, works with powers.

Knowing how to apply these tests helps to analyze the behavior of infinite series and understand their properties.

An alternating series is a series in which the terms alternate in sign. This can be seen in series where the terms contain factors like \( (-1)^n \) or \( \cos(n\pi) \), which alternate between positive and negative as \( n \) increases. Alternating series often take the general form:

• \( a_1 - a_2 + a_3 - a_4 + \ldots \)

The alternating series have their own test for convergence known as the Alternating Series Test. According to this test, an alternating series \( \sum (-1)^n a_n \) converges if:

• The absolute values of the terms \( a_n \) decrease monotonically: \( a_{n+1} \leq a_n \) for all \( n \).
• The terms approach zero: \( \lim_{{n \to \infty}} a_n = 0 \).

If both conditions are satisfied, the series converges, even though it oscillates indefinitely. Understanding how alternating series behave is important to correctly identifying their convergence characteristics.