Which Series Has A Sum Of 5?A. $\sum_{n=1}^{\infty} 2(0.1)^{n-1}$ B. $\sum_{n=1}^{\infty} 15(0.3)^{n-1}$ C. $\sum_{n=1}^{\infty} 4(0.2)^{n-1}$ D. $\sum_{n=1}^{\infty} 20(0.4)^{n-1}$

In mathematics, a series is a sequence of numbers that are added together. The sum of a series is the total value of all the numbers in the series. In this article, we will explore four different series and determine which one has a sum of 5.

Understanding the Series

Before we dive into the calculations, let's understand the general form of a series. A series can be represented as:

$$\sum_{n=1}^{\infty} a_n$$

where $a_n$ is the nth term of the series. In the context of the given problem, we have four different series, each with its own formula:

• A: $\sum_{n=1}^{\infty} 2(0.1)^{n-1}$
• B: $\sum_{n=1}^{\infty} 15(0.3)^{n-1}$
• C: $\sum_{n=1}^{\infty} 4(0.2)^{n-1}$
• D: $\sum_{n=1}^{\infty} 20(0.4)^{n-1}$

Calculating the Sum of Each Series

To determine which series has a sum of 5, we need to calculate the sum of each series. We can use the formula for the sum of an infinite geometric series:

$$S = \frac{a}{1 - r}$$

where $a$ is the first term of the series and $r$ is the common ratio.

Series A

For series A, the first term $a$ is 2 and the common ratio $r$ is 0.1. Plugging these values into the formula, we get:

$$S_A = \frac{2}{1 - 0.1} = \frac{2}{0.9} = \frac{20}{9}$$

Series B

For series B, the first term $a$ is 15 and the common ratio $r$ is 0.3. Plugging these values into the formula, we get:

$$S_B = \frac{15}{1 - 0.3} = \frac{15}{0.7} = \frac{150}{7}$$

Series C

For series C, the first term $a$ is 4 and the common ratio $r$ is 0.2. Plugging these values into the formula, we get:

$$S_C = \frac{4}{1 - 0.2} = \frac{4}{0.8} = 5$$

Series D

For series D, the first term $a$ is 20 and the common ratio $r$ is 0.4. Plugging these values into the formula, we get:

$$S_D = \frac{20}{1 - 0.4} = \frac{20}{0.6} = \frac{100}{3}$$

Conclusion

Based on the calculations, we can see that series C has a sum of 5. This is because the common ratio $r$ is 0.2, which is less than 1, and the first term $a$ is 4, which is a positive number. As a result, the sum of the series is 5.

Comparison of the Series

In this article, we compared four different series and determined which one has a sum of 5. We used the formula for the sum of an infinite geometric series to calculate the sum of each series. The results show that series C has a sum of 5, while the other series have sums that are not equal to 5.

Key Takeaways

• A series is a sequence of numbers that are added together.
• The sum of a series is the total value of all the numbers in the series.
• The formula for the sum of an infinite geometric series is $S = \frac{a}{1 - r}$.
• Series C has a sum of 5, while the other series have sums that are not equal to 5.

Real-World Applications

The concept of series and their sums has many real-world applications. For example, in finance, the sum of a series can be used to calculate the present value of a future stream of payments. In engineering, the sum of a series can be used to calculate the total stress on a structure.

Future Research Directions

There are many areas of research that involve the study of series and their sums. Some potential areas of research include:

• Developing new formulas for the sum of series.
• Investigating the convergence of series.
• Applying series to real-world problems.

Conclusion

In the previous article, we explored the concept of series and their sums. In this article, we will answer some frequently asked questions about series and their sums.

Q: What is a series?

A: A series is a sequence of numbers that are added together. It is a way of representing a sum of an infinite number of terms.

Q: What is the sum of a series?

A: The sum of a series is the total value of all the numbers in the series. It is the result of adding up all the terms in the series.

Q: How do I calculate the sum of a series?

A: To calculate the sum of a series, you can use the formula for the sum of an infinite geometric series:

$$S = \frac{a}{1 - r}$$

where $a$ is the first term of the series and $r$ is the common ratio.

Q: What is the common ratio?

A: The common ratio is the ratio of each term to the previous term in the series. It is a number that is multiplied by the previous term to get the next term.

Q: How do I determine if a series converges?

A: A series converges if the sum of the series is finite. In other words, if the series has a sum that is not infinite, then it converges.

Q: What is the difference between a convergent and a divergent series?

A: A convergent series is a series that has a finite sum. A divergent series is a series that has an infinite sum.

Q: Can a series have both convergent and divergent terms?

A: Yes, a series can have both convergent and divergent terms. However, the series as a whole will only converge if the sum of the convergent terms is finite.

Q: How do I apply series to real-world problems?

A: Series can be applied to a wide range of real-world problems, including finance, engineering, and economics. For example, in finance, the sum of a series can be used to calculate the present value of a future stream of payments.

Q: What are some common types of series?

A: Some common types of series include:

• Geometric series: A series in which each term is a constant multiple of the previous term.
• Arithmetic series: A series in which each term is a constant difference from the previous term.
• Harmonic series: A series in which each term is the reciprocal of the previous term.

Q: What are some common applications of series?

A: Some common applications of series include:

• Finance: Series can be used to calculate the present value of a future stream of payments.
• Engineering: Series can be used to calculate the total stress on a structure.
• Economics: Series can be used to model economic systems and make predictions about future economic trends.

Conclusion

In conclusion, series and their sums are an important concept in mathematics that has many real-world applications. By understanding the formula for the sum of an infinite geometric series and the concept of convergence, you can apply series to a wide range of problems.

Frequently Asked Questions

• What is a series?
• What is the sum of a series?
• How do I calculate the sum of a series?
• What is the common ratio?
• How do I determine if a series converges?
• What is the difference between a convergent and a divergent series?
• Can a series have both convergent and divergent terms?
• How do I apply series to real-world problems?
• What are some common types of series?
• What are some common applications of series?

Glossary

• Series: A sequence of numbers that are added together.
• Sum: The total value of all the numbers in the series.
• Common ratio: The ratio of each term to the previous term in the series.
• Convergent series: A series that has a finite sum.
• Divergent series: A series that has an infinite sum.
• Geometric series: A series in which each term is a constant multiple of the previous term.
• Arithmetic series: A series in which each term is a constant difference from the previous term.
• Harmonic series: A series in which each term is the reciprocal of the previous term.