An alternating series is a series in which the terms alternate in sign, meaning that every other term is positive while the rest are negative. This type of series is used in mathematical and statistical analysis to approximate values or to test for convergence.
Definition:
An alternating series is a series of the form:
a_1 – a_2 + a_3 – a_4 + a_5 – …
where each term, a_i, is positive or negative. The series is said to converge if the sum of the terms approaches a specific value as the number of terms increases. The value to which the series converges is called the sum of the series.
Examples:
- The series 1 – 1/3 + 1/5 – 1/7 + 1/9 – … is an alternating series. This series is a common example used to demonstrate the convergence of an alternating series. The sum of this series is approximately 0.68, which can be approximated using a partial sum formula or by using a calculator or computer program.
- The series 1 – 1/4 + 1/9 – 1/16 + 1/25 – … is an alternating series. This series is a geometric series with a common ratio of -1/4. The sum of this series is approximately 0.84, which can be approximated using a partial sum formula or by using a calculator or computer program.
- The series 1 – 1/9 + 1/25 – 1/49 + 1/81 – … is an alternating series. This series is a geometric series with a common ratio of -1/9. The sum of this series is approximately 0.92, which can be approximated using a partial sum formula or by using a calculator or computer program.
- The series 1 – 1/8 + 1/27 – 1/64 + 1/125 – … is an alternating series. This series is a geometric series with a common ratio of -1/8. The sum of this series is approximately 0.94, which can be approximated using a partial sum formula or by using a calculator or computer program.
- The series 1 – 1/5 + 1/14 – 1/41 + 1/122 – … is an alternating series. This series is a geometric series with a common ratio of -1/5. The sum of this series is approximately 0.96, which can be approximated using a partial sum formula or by using a calculator or computer program.
Quiz:
- What is an alternating series?
- What is the sum of an alternating series?
- What is the formula for determining the sum of an alternating series?
- Is the series 1 – 1/3 + 1/5 – 1/7 + 1/9 – … an alternating series?
- Is the series 1 + 1/3 – 1/5 + 1/7 – 1/9 + … an alternating series?
- Is the series 1 – 1/4 + 1/9 – 1/16 + 1/25 – … a geometric series?
- Is the series 1 – 1/9 + 1/25 – 1/49 + 1/81 – … a geometric series?
- Is the series 1 – 1/8 + 1/27 – 1/64 + 1/125 – … a geometric series?
- Is the series 1 – 1/5 + 1/14 – 1/41 + 1/122 – … a geometric series?
- Can the sum of an alternating series be approximated using a partial sum
