Find the sum of each infinite geometric series. 2 - 1 + 1/2 - 1/4 + ...

Verified step by step guidance

1. Identify the first term $a$ of the series, which is 2.
2. Determine the common ratio $r$ by dividing the second term by the first term: $r=\frac{-1}{2}$.
3. Check that the series is geometric and that the absolute value of the common ratio $|r|<1$ to ensure the series converges.
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 to find the sum.

Key Concepts

Infinite Geometric Series

An infinite geometric series is a sum of the terms of a geometric sequence that continues indefinitely. It is defined by a first term 'a' and a common ratio 'r'. The series converges to a finite value if the absolute value of the common ratio is less than one (|r| < 1). The formula for the sum of an infinite geometric series is S = a / (1 - r).

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to obtain the next term. It is calculated by dividing any term by its preceding term. For the series 2, -1, 1/2, -1/4, the common ratio can be found by taking -1/2 divided by 2, which equals -1/2. This ratio is crucial for determining the convergence of the series.

Convergence of Series

Convergence refers to the behavior of a series as the number of terms approaches infinity. An infinite series converges if the sum approaches a specific finite value. For geometric series, convergence occurs when the absolute value of the common ratio is less than one. If the series diverges, it means the sum does not approach a finite limit, which is essential to assess when calculating the sum of an infinite series.