In Exercises 37–44, find the sum of each infinite geometric series. 3 + 3/4 + 3/4^2 + 3/4^3 + ...

Verified step by step guidance

1. Identify the first term $a$ of the series, which is 3.
2. Determine the common ratio $r$ by dividing the second term by the first term: $r=\frac{3/4}{3}=\frac{1}{4}$.
3. Verify that the series is geometric and that the common ratio $|r|<1$, which is true since $\frac{1}{4}<1$.
4. Use the formula for the sum of an infinite geometric series: $S=\frac{a}{1-r}$.
5. Substitute the values of $a$ and $r$ into the formula: $S=\frac{3}{1-\frac{1}{4}}$.

Key Concepts

Infinite Geometric Series

An infinite geometric series is a sum of the form a + ar + ar^2 + ar^3 + ... where 'a' is the first term and 'r' is the common ratio. This series continues indefinitely. The series converges (has a finite sum) if the absolute value of the common ratio 'r' is less than 1.

Sum of an Infinite Geometric Series

The sum S of an infinite geometric series can be calculated using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. This formula is applicable only when |r| < 1, ensuring that the series converges to a finite value.

Convergence of Series

Convergence refers to the behavior of a series as the number of terms approaches infinity. For an infinite geometric series, convergence occurs when the common ratio 'r' is between -1 and 1. If 'r' is outside this range, the series diverges, meaning it does not approach a finite limit.