The sum of an infinite geometric series, represented by the formula S = a/(1-r), is an important concept in mathematics with applications in areas such as finance, probability, and physics. Its four closely related entities include the first term (a), the common ratio (r), the number of terms (n), and the sum (S). The first term represents the initial value of the series, while the common ratio determines the rate of change between consecutive terms. The number of terms indicates the number of elements in the series, and the sum is the total value of the series.
Structure for Summing Up an Infinite Geometric Series
The sum of an infinite geometric series is given by the formula:
S = a / (1 - r)
where:
ais the first term in the seriesris the common ratio (the ratio of any two consecutive terms in the series)
For example, the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ... is:
S = 1 / (1 - 1/2) = 2
Here are some important points to keep in mind when using this formula:
- The series must be convergent, which means that
|r| < 1. Otherwise, the sum will be infinite. - The formula is only valid for infinite geometric series. For finite geometric series, you can use the formula:
S = a(1 - r^n) / (1 - r)
where n is the number of terms in the series.
Here is a table summarizing the key information about the sum of an infinite geometric series:
| Property | Description |
|---|---|
| Formula | S = a / (1 - r) |
| Convergence | The series must be convergent, which means that |r| < 1. |
| Finite series | For finite geometric series, use the formula: S = a(1 - r^n) / (1 - r) |
Question 1:
What is the sum of an infinite geometric series?
Answer:
The sum of an infinite geometric series with first term a and common ratio r, where |r| < 1, is given by S = a / (1 - r).
Question 2:
Under what conditions does an infinite geometric series exist?
Answer:
An infinite geometric series exists only when the absolute value of the common ratio r is less than 1. If |r| ≥ 1, the series diverges.
Question 3:
How can you use the sum of an infinite geometric series to find the value of an annuity?
Answer:
To find the value of an annuity, you can use the sum of an infinite geometric series with first term equal to the present value of the first payment and common ratio equal to the interest rate per period.
Well, there you have it, folks! I hope this little trip into the world of infinite geometric series has been illuminating. Remember, the sum of an infinite geometric series is a powerful concept with many applications in the real world. From calculating the total cost of a loan to predicting the spread of a virus, this formula can be a valuable tool. Thanks for reading, and be sure to check back for more math adventures in the future!