Problem 40 - College Algebra

Question

Find the sum of each infinite geometric series. $$5+\frac{5}{6}+\frac{5}{6^{2}}+\frac{5}{6^{3}}+\cdots$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is 6.

Step 1: Identify the first term and the ratio

The first term \(a\) of the series is 5 and the ratio \(r\) is \(\frac{5}{6} / 5 = \frac{1}{6}\). Now, we can see that the absolute value of the ratio is less than 1, which means the series is convergent and we may proceed with using the formula for the sum of an infinite series.

Step 2: Apply the formula for the sum of a geometric series

Insert the values of \(a\) and \(r\) into the formula: \(S = \frac{a}{1 - r} = \frac{5}{1 - \frac{1}{6}}\).

Step 3: Simplifying

Simplify the expression to get: \(S = \frac{5}{\frac{5}{6}} = 6\).