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An infinite series is essentially a sum of infinitely many terms. When dealing with sequences and series, particularly those of infinite length, it's crucial to understand what these terms represent and how they are summed. An infinite series is generally represented in the form \( \sum_{k=1}^{\infty} a_k \), where \( a_k \) is the individual terms of the sequence.
A series can sometimes be extremely complex depending on the terms involved, but often in mathematics, we are interested in whether the sum approaches a specific value as more terms are added. This is where concepts like convergence come into play, helping us determine if the series can be assigned a finite sum value despite having infinitely many terms. For geometric series, which are a special type of infinite series, they converge under certain conditions. Let's explore this more in the next section.

For an infinite series to be meaningful, mathematicians need to discuss its convergence. Convergence refers to the idea that as we keep adding terms from a series, the sum approaches a specific, fixed number. If the terms of a series do not approach a specific limit, the series is said to be divergent.

• A series converges if the sequence of its partial sums tends to a limit, as the number of terms goes toward infinity.
• For geometric series, convergence is straightforward to check: If the absolute value of the common ratio \(|r|\) is less than 1, then the series converges. Otherwise, it diverges.
• In our original exercise, both series had common ratios of \(\frac{5}{6}\) and \(\frac{7}{9}\), which are less than 1, confirming that each series converges.

Understanding convergence is key when working with infinite series because it informs us whether we can meaningfully sum the series to get a finite result. Convergent series are not only meaningful but are also manageable with specific formulas to find their sums, which we'll see in the final section.

When dealing with geometric series, there's a very handy formula to find their sum, as long as the series converges. For a geometric series with a first term \(a\) and a common ratio \(r\), the sum of the infinite series is given by:
\[S = \frac{a}{1 - r}\]This formula arises from the property that the series terms multiply over each other in a consistent manner, which makes it easier to handle them in a summation context.

• The first term \(a\) is the term with which the sequence begins.
• The formula is applicable, provided \(|r|<1\), ensuring convergence.
• Using this formula, we calculated the sums of each component series separately in our original problem to reach a final result of \(\frac{47}{10}\).

This method of using the geometric series sum formula simplifies the computation significantly and is a powerful tool when addressing infinite sums of geometric sequences. It's important to confirm that the common ratio condition holds before applying the formula to ensure a valid result.