The sum of an infinite geometric series with first term \( a = 5 \) and common ratio \( r = \frac{1}{3} \) is given by \( S = \frac{a}{1 - r} \). Calculate \( S \).
["The Sum of an Infinite Geometric Series: A Clear Guide with Example", "Understanding infinite series is a fundamental concept in mathematics, especially in calculus, finance, and engineering. One of the most elegant and widely used formulas involves the sum of an infinite geometric series. This article explains the formula, its application, and walks through a concrete example using specific values: first term ( a = 5 ) and common ratio ( r = \frac{1}{3} ).", "---", "### What is an Infinite Geometric Series?", "A geometric series is a sequence where each term is multiplied by a constant ratio to get the next term. For an infinite geometric series, the pattern continues forever:", "[\nS = a + ar + ar^2 + ar^3 + \cdots\n]", "where:\n- ( a ) is the first term,\n- ( r ) is the common ratio.", "This sum converges (i.e., reaches a finite value) only when ( |r| < 1 ). If ( |r| \geq 1 ), the series diverges and the sum is undefined.", "---", "### The Formula for the Sum", "For a convergent geometric series (( |r| < 1 )), the sum ( S ) is given by:", "[\nS = \frac{a}{1 - r}\n]", "This formula provides a quick way to compute the total of an infinite sequence without summing each term individually.", "---", "### Applying the Formula: Step-by-Step Example", "Let’s apply this concept using the values ( a = 5 ) and ( r = \frac{1}{3} ).", "1. Check convergence condition:\n Since ( |r| = \left| \frac{1}{3} \right| = \frac{1}{3} < 1 ), the series converges, and the sum exists.", "2. Substitute into the formula:", "[\nS = \frac{a}{1 - r} = \frac{5}{1 - \frac{1}{3}}\n]", "3. Simplify the denominator:", "[\n1 - \frac{1}{3} = \frac{2}{3}\n]", "4. Compute the sum:", "[\nS = \frac{5}{\frac{2}{3}} = 5 \ imes \frac{3}{2} = \frac{15}{2} = 7.5\n]", "---", "### Final Result", "The sum of the infinite geometric series with ( a = 5 ) and ( r = \frac{1}{3} ) is:", "[\n\boxed{S = \frac{15}{2}} \quad \ ext{or} \quad \boxed{7.5}\n]", "---", "### Why This Formula Matters", "This principle is essential in practical fields such as finance (calculating present values of perpetual annuities), physics (modeling damped oscillations), and computer science (analyzing recursive algorithms). Mastering it provides a powerful tool for understanding infinite processes efficiently.", "If you're working with geometric series, always verify convergence first—knowing when the sum formula applies makes all the difference.", "---", "Keywords: infinite geometric series sum, formula for sum infinite geometric series, ( S = \frac{a}{1 - r} ), convergent geometric series, mathematical formula explanation, calculate infinite series sum.\nMeta Description: Learn how to calculate the sum of an infinite geometric series with values ( a = 5 ) and ( r = \frac{1}{3} ) using the formula ( S = \frac{a}{1 - r} ). Step-by-step solution and real-world relevance explained."]
