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A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In mathematical terms, a geometric series with first term \( a \) and common ratio \( r \) is

• \( a, ar, ar^2, ar^3, \ldots \)

The series can be summed up, forming a infinite series if \( |r| < 1 \), meaning the terms get smaller and smaller, approaching zero. This is key in many mathematical calculations because it allows infinite series to be summed to a finite value. In the original exercise, each series part \( 2^{-n} \) and \( 3^{-n} \) follows a geometric sequence, with respective common ratios \( \frac{1}{2} \) and \( \frac{1}{3} \). Understanding geometric series is crucial for tackling problems involving exponential decay or growth, as these often have terms that behave in a similar way.

Convergence in mathematical series refers to whether an infinite series approaches a specific value as more terms are added. For a geometric series specifically, convergence is guaranteed when the common ratio \( r \) satisfies \( |r| < 1 \). This means that each subsequent term becomes progressively smaller, ensuring the sum doesn't spiral towards infinity.
In the exercise at hand, each split series \( \sum_{n=3}^{\infty} 2^{-n} \) and \( \sum_{n=3}^{\infty} 3^{-n} \) individually converge. Here, \( 2^{-n} \) with a ratio \( \frac{1}{2} \), and \( 3^{-n} \) with a ratio \( \frac{1}{3} \), both fall well within the \( |r| < 1 \) requirement, validating their convergence. Recognizing convergence is fundamental when dealing with series, as it tells us that the series does not differ indefinitely, rather it approaches a specific finite number allowing practical calculation of sums.

Calculating the sum of an infinite series involves using specific formulas designed for that type of series. For a geometric series, the sum \( S \) can be evaluated from the formula

• \( S = \frac{a}{1-r} \)

where \( a \) is the first term and \( r \) is the common ratio. This formula applies when \( |r| < 1 \), ensuring that the series converges to a finite number. In the original exercise, we see this formula in action as each sub-series is calculated separately.
For \( \sum_{n=3}^{\infty} 2^{-n} \), the first term \( a = \frac{1}{8} \) gives a sum of \( \frac{1}{4} \). Similarly, for \( \sum_{n=3}^{\infty} 3^{-n} \), \( a = \frac{1}{27} \) results in a sum of \( \frac{1}{18} \). Once calculated, these sums are then added to find the total sum of the series, which demonstrates the practical application of understanding and working with infinite series in mathematical exercises.