Applying the Formula for Infinite Geometric Series

Oct 13, 2025 by windailys.com 34 views

Understanding the Formula for Infinite Geometric Series

In mathematics, an infinite geometric series is a series of the form ${\sum_{k=0}^{\infty} ar^k}$ , where ${a}$ is the first term and ${r}$ is the common ratio. The sum of an infinite geometric series can be calculated using the formula ${S_{\infty} = \frac{a_1}{1-r}}$ , where ${0 < |r| < 1}$ . This formula is essential in various mathematical and real-world applications, including finance, engineering, and physics.

The Given Problem: Finding the Sum of an Infinite Geometric Series

We are given the infinite geometric series ${\sum_{k=0}^{\infty} 5\left(\frac{3}{4}\right)^k}$ . To find the sum of this series, we need to apply the formula for infinite geometric series. The first term ${a_1}$ is 5, and the common ratio ${r}$ is ${\frac{3}{4}}$ .

Applying the Formula for Infinite Geometric Series

To find the sum of the infinite geometric series, we substitute the values of ${a_1}$ and ${r}$ into the formula ${S_{\infty} = \frac{a_1}{1-r}}$ . This gives us:

${S_{\infty} = \frac{5}{1-\frac{3}{4}}}$

Simplifying the Expression

To simplify the expression, we first simplify the denominator:

${1-\frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4}}$

Now, we substitute this value back into the expression:

${S_{\infty} = \frac{5}{\frac{1}{4}}}$

Further Simplification

To further simplify the expression, we can multiply the numerator by 4 to eliminate the fraction in the denominator:

${S_{\infty} = 5 \times 4 = 20}$

Conclusion

Therefore, the sum of the infinite geometric series ${\sum_{k=0}^{\infty} 5\left(\frac{3}{4}\right)^k}$ is 20.

Real-World Applications of Infinite Geometric Series

Infinite geometric series have numerous real-world applications, including:

Finance: Infinite geometric series can be used to calculate the present value of an annuity, which is a series of equal payments made at regular intervals.
Engineering: Infinite geometric series can be used to calculate the sum of an infinite series of resistors or capacitors in an electrical circuit.
Physics: Infinite geometric series can be used to calculate the sum of an infinite series of waves or vibrations in a physical system.

Common Mistakes to Avoid

When working with infinite geometric series, there are several common mistakes to avoid:

Incorrect calculation of the sum: Make sure to apply the formula correctly and simplify the expression carefully.
Incorrect identification of the first term and common ratio: Make sure to identify the first term and common ratio correctly before applying the formula.
Incorrect assumption about the convergence of the series: Make sure to check that the series converges before applying the formula.

Conclusion

In conclusion, infinite geometric series are a fundamental concept in mathematics, and the formula for calculating their sum is essential in various mathematical and real-world applications. By understanding the formula and applying it correctly, we can solve problems involving infinite geometric series and make informed decisions in finance, engineering, and physics.

Q: What is an infinite geometric series?

A: An infinite geometric series is a series of the form ${\sum_{k=0}^{\infty} ar^k}$ , where ${a}$ is the first term and ${r}$ is the common ratio. The series continues indefinitely, with each term being obtained by multiplying the previous term by the common ratio.

Q: What is the formula for the sum of an infinite geometric series?

A: The formula for the sum of an infinite geometric series is ${S_{\infty} = \frac{a_1}{1-r}}$ , where ${a_1}$ is the first term and ${r}$ is the common ratio. This formula is valid only when ${0 < |r| < 1}$ .

Q: What is the condition for convergence of an infinite geometric series?

A: An infinite geometric series converges only when ${0 < |r| < 1}$ . If ${|r| \geq 1}$ , the series diverges.

Q: How do I determine whether an infinite geometric series converges or diverges?

A: To determine whether an infinite geometric series converges or diverges, you need to check the value of the common ratio ${r}$ . If ${0 < |r| < 1}$ , the series converges. If ${|r| \geq 1}$ , the series diverges.

Q: What is the sum of the infinite geometric series ${\sum_{k=0}^{\infty} 2\left(\frac{1}{3}\right)^k}$ ?

A: To find the sum of this series, we apply the formula for infinite geometric series:

${S_{\infty} = \frac{a_1}{1-r} = \frac{2}{1-\frac{1}{3}} = \frac{2}{\frac{2}{3}} = 3}$

Q: What is the sum of the infinite geometric series ${\sum_{k=0}^{\infty} 3\left(\frac{2}{5}\right)^k}$ ?

A: To find the sum of this series, we apply the formula for infinite geometric series:

${S_{\infty} = \frac{a_1}{1-r} = \frac{3}{1-\frac{2}{5}} = \frac{3}{\frac{3}{5}} = 5}$

Q: What is the sum of the infinite geometric series ${\sum_{k=0}^{\infty} 4\left(\frac{3}{2}\right)^k}$ ?

A: To find the sum of this series, we apply the formula for infinite geometric series:

${S_{\infty} = \frac{a_1}{1-r} = \frac{4}{1-\frac{3}{2}} = \frac{4}{\frac{-1}{2}} = -8}$

Q: Can I use the formula for infinite geometric series to find the sum of a finite geometric series?

A: No, the formula for infinite geometric series is only valid for infinite series. To find the sum of a finite geometric series, you need to use a different formula.

Q: What is the difference between an infinite geometric series and a finite geometric series?

A: An infinite geometric series is a series that continues indefinitely, while a finite geometric series is a series that has a finite number of terms.

Q: Can I use the formula for infinite geometric series to find the sum of a series with a common ratio of 1?

A: No, the formula for infinite geometric series is only valid when ${0 < |r| < 1}$ . If ${|r| \geq 1}$ , the series diverges.

Q: Can I use the formula for infinite geometric series to find the sum of a series with a common ratio of -1?

A: No, the formula for infinite geometric series is only valid when ${0 < |r| < 1}$ . If ${|r| \geq 1}$ , the series diverges.

Q: Can I use the formula for infinite geometric series to find the sum of a series with a common ratio of 0?

A: No, the formula for infinite geometric series is only valid when ${0 < |r| < 1}$ . If ${r = 0}$ , the series is a constant series and does not converge to a finite sum.