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An infinite series is a sum that continues indefinitely, often represented as the sum of an infinite number of terms. A classic example is the geometric series, where each term is a constant multiple of the previous one. Unlike finite sums, which add up a set number of terms, infinite series continue without bound. However, this does not always mean they add up to an infinite value. Even without a last term, many infinite series can have well-defined sums. This is where the idea of convergence comes into play, determining whether the sum of the series can reach a specific finite number.

• Each term in the series contributes a portion to the overall sum.
• The series examined in our context is vast, starting from a specific term and extending to infinity.

For example, in our given series from the exercise, the terms are of the form \( \frac{1}{5^{k}} \) which starts at \( k=4 \) and stretches out to infinity.

Convergence is a vital concept when dealing with an infinite series. The notion of convergence answers the question of whether an infinite series settles at a particular value as more terms are considered. A series is said to be convergent if, as you add more and more terms, the series approaches a finite limit. The key determinant of convergence in a geometric series is the common ratio \( r \).

• If the absolute value of \( r \) is less than 1, the series converges.
• If \( |r| \geq 1 \), the series diverges, meaning it doesn't settle at a finite value.

In the example given, the common ratio \( r = \frac{1}{5} \) is indeed less than 1, ensuring that the series converges. Hence, this means that the sum of the series does not stretch to infinity but resolves to a specific number, which helps in making the series manageable and calculable.

Once the convergence of a geometric series is established, we can compute its sum using a straightforward formula. For a convergent geometric series with a first term \( a \) and common ratio \( r \), the sum \( S_\infty \) can be calculated as:\[ S_\infty = \frac{a}{1 - r} \]This formula provides the means to find the total sum regardless of the infinite number of terms, as long as the series converges. Calculating the actual sum involves substituting the known values of \( a \) and \( r \) into the formula. In our exercise:

• The first term \( a \) is \( \frac{1}{625} \) (found by setting \( k=4\) in the formula \( \frac{1}{5^{k}} \)).
• The common ratio \( r \) is \( \frac{1}{5} \).

These values plugged into the sum formula yield \( S_\infty = \frac{1}{500} \), demonstrating how the series, despite being infinite, equates to a simple finite number.