Find the sum of the infinite geometric series with first term 5 and common ratio \( \frac{2}{3} \).
["Title: How to Find the Sum of an Infinite Geometric Series: Example with First Term 5 and Ratio ( \frac{2}{3} )", "When working with infinite sequences, one powerful mathematical tool is the infinite geometric series. Infinite geometric series appear in finance, engineering, and computer science, making understanding their sum essential. In this article, we explore how to find the sum of an infinite geometric series with a first term of ( 5 ) and a common ratio of ( \frac{2}{3} ).", "### What Is an Infinite Geometric Series?", "An infinite geometric series is the sum of infinitely many terms where each term is a constant multiple (called the common ratio) of the previous term. It has the form:\n[\nS = a + ar + ar^2 + ar^3 + \cdots\n]\nwhere:\n- ( a ) = first term\n- ( r ) = common ratio, with ( |r| < 1 ) (absolute value less than one for convergence)", "### Conditions for Convergence", "A crucial requirement for the sum of an infinite geometric series to exist is that the absolute value of the common ratio must be less than 1:\n[\n|r| < 1\n]\nIn our example, the common ratio is ( r = \frac{2}{3} ), which satisfies ( |r| = \frac{2}{3} < 1 ). Therefore, the series converges, and the sum can be computed using a clear formula.", "### The Formula for the Sum", "The sum ( S ) of an infinite geometric series is given by:\n[\nS = \frac{a}{1 - r}, \quad \ ext{for } |r| < 1\n]", "### Plugging in the Values", "Here, ( a = 5 ) and ( r = \frac{2}{3} ). Substitute these into the formula:\n[\nS = \frac{5}{1 - \frac{2}{3}}\n]", "First, simplify the denominator:\n[\n1 - \frac{2}{3} = \frac{1}{3}\n]", "Now compute the sum:\n[\nS = \frac{5}{\frac{1}{3}} = 5 \ imes 3 = 15\n]", "### Conclusion", "The sum of the infinite geometric series with first term ( 5 ) and common ratio ( \frac{2}{3} ) is ( 15 ). This result is derived from the fundamental formula, valid only because ( \left| \frac{2}{3} \right| < 1 ), ensuring convergence.", "Understanding this concept not only helps solve mathematical problems elegantly but also provides practical insight into compound growth, present value calculations, and series modeling—making it a vital skill in both academic and real-world applications.", "---", "Key SEO Keywords: infinite geometric series, sum of geometric series, find sum infinite series, formula convergence condition, mathematical series example, infinite series sum calculation", "Meta Description:\nLearn how to find the sum of an infinite geometric series with first term 5 and common ratio ( \frac{2}{3} ). Step-by-step explanation, formula, and convergence proof included."]
