91Ó°ÊÓ

Convergence in mathematics refers to the behavior of a series as it approaches a specific value. For a geometric series, convergence is determined mainly by the common ratio, denoted as \(r\). Given a series \( \sum a_k \), if the limit as \(k\) approaches infinity is finite, the series converges. In the context of geometric series, convergence primarily depends on the comparison of the absolute value of \(r\) to 1. A series converges if

• \(|r| < 1\)

This criterion tells us that the terms of the series are getting smaller and approaching zero, allowing the entire series to sum up to a specific limit. In the provided exercise, since the common ratio is \(r = \frac{1}{5}\), convergence is assured. This is because \(\left| \frac{1}{5} \right| = 0.2\), which is indeed less than 1, confirming that the series is converging to a definite sum.

An infinite series is essentially a summation of a sequence of terms that continues indefinitely. When we talk about an infinite geometric series, we mean a series of the form \( \sum_{k=1}^{\infty} ar^{k-1} \) where the terms multiply by a constant ratio \(r\). The beauty of infinite series lies in their behavior and the fascinating ways they can sum to finite values, despite having an infinite number of terms. In practical cases, these series help in solving real-world problems including those in finance, physics, and engineering.Our exercise deals with an infinite series starting from \(k=4\). Infinite series such as these converge only if their common ratio, \(|r|\), is less than 1. This gives the series a ceiling, a maximum limit it cannot exceed, thus summing into a finite number. The understanding of infinite series plays a crucial role in determining the bounds and behavior of mathematical functions.

The sum formula for an infinite geometric series provides us with a straightforward way to find the total of all terms in the series. When a geometric series converges, as determined by its common ratio \(|r| < 1\), its sum can be calculated with the formula: \[ S = \frac{a_1}{1 - r} \]where \(S\) represents the sum, \(a_1\) is the first term of the series, and \(r\) is the common ratio.In our exercise, to find the sum of the series \( \sum_{k=4}^{\infty} \frac{1}{5^{k}} \), the first term was calculated as \(\frac{1}{625}\), with the common ratio as \(\frac{1}{5}\). Plugging these values into the formula yields:

• \( S = \frac{\frac{1}{625}}{1 - \frac{1}{5}} = \frac{1}{625} \times \frac{5}{4} \)

The result is \(\frac{5}{2500}\), which is the sum of the infinite series. This simple yet profound formula allows us to harness the power of infinite processes to deliver tangible, finite results.