Understanding the Infinite Series

Oct 13, 2025 by windailys.com 115 views

Solving for the Value of c in an Infinite Series

In mathematics, infinite series are a fundamental concept used to represent the sum of an infinite number of terms. These series can be used to model real-world phenomena, such as population growth, electrical circuits, and more. In this article, we will explore the value of c in an infinite series, given the equation ∑n=1 to ∞^-n = 6, where c > 0.

Understanding the Infinite Series

The given infinite series is a geometric series, which is a series of the form ∑[n=1 to ∞]ar^(n-1), where a is the first term and r is the common ratio. In this case, the first term is 1 and the common ratio is (1+c)^-1. The sum of an infinite geometric series can be calculated using the formula S = a / (1 - r), where S is the sum of the series.

Applying the Formula for the Sum of an Infinite Geometric Series

Using the formula for the sum of an infinite geometric series, we can rewrite the given equation as:

∑n=1 to ∞^-n = 1 / (1 - (1+c)^-1) = 6

Simplifying the Equation

To simplify the equation, we can start by multiplying both sides by (1 - (1+c)^-1):

1 = 6(1 - (1+c)^-1)

Expanding and Simplifying

Expanding the right-hand side of the equation, we get:

1 = 6 - 6(1+c)^-1

Subtracting 6 from both sides, we get:

-5 = -6(1+c)^-1

Multiplying both sides by -1, we get:

5 = 6(1+c)^-1

Solving for c

To solve for c, we can start by multiplying both sides of the equation by (1+c):

5(1+c) = 6

Expanding the left-hand side of the equation, we get:

5 + 5c = 6

Subtracting 5 from both sides, we get:

5c = 1

Dividing both sides by 5, we get:

c = 1/5

In this article, we have solved for the value of c in an infinite series, given the equation ∑n=1 to ∞^-n = 6, where c > 0. Using the formula for the sum of an infinite geometric series, we were able to simplify the equation and solve for c. The value of c is 1/5.

The final answer is: $\boxed{\frac{1}{5}}$
Q&A: Solving for the Value of c in an Infinite Series

In our previous article, we explored the value of c in an infinite series, given the equation ∑n=1 to ∞^-n = 6, where c > 0. We used the formula for the sum of an infinite geometric series to simplify the equation and solve for c. In this article, we will answer some frequently asked questions about solving for the value of c in an infinite series.

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 = a / (1 - r), where S is the sum of the series, a is the first term, and r is the common ratio.

Q: How do I know if an infinite series is a geometric series?

A: An infinite series is a geometric series if it can be written in the form ∑[n=1 to ∞]ar^(n-1), where a is the first term and r is the common ratio.

Q: What is the common ratio in the given infinite series?

A: The common ratio in the given infinite series is (1+c)^-1.

Q: How do I simplify the equation ∑n=1 to ∞^-n = 6?

A: To simplify the equation, you can use the formula for the sum of an infinite geometric series. Multiply both sides of the equation by (1 - (1+c)^-1) to get 1 = 6(1 - (1+c)^-1).

Q: How do I solve for c in the equation 5(1+c) = 6?

A: To solve for c, you can start by expanding the left-hand side of the equation to get 5 + 5c = 6. Then, subtract 5 from both sides to get 5c = 1. Finally, divide both sides by 5 to get c = 1/5.

Q: What is the value of c in the given infinite series?

A: The value of c in the given infinite series is 1/5.

Q: Can I use the same method to solve for c in other infinite series?

A: Yes, you can use the same method to solve for c in other infinite series that are geometric series. However, you may need to adjust the formula for the sum of an infinite geometric series to fit the specific series you are working with.

Q: What are some common mistakes to avoid when solving for c in an infinite series?

A: Some common mistakes to avoid when solving for c in an infinite series include:

Not checking if the series is a geometric series before applying the formula for the sum of an infinite geometric series.
Not simplifying the equation correctly before solving for c.
Not checking if the value of c is valid (e.g. c > 0).

In this article, we have answered some frequently asked questions about solving for the value of c in an infinite series. We have also provided some tips and common mistakes to avoid when working with infinite series. By following these tips and avoiding common mistakes, you can become more confident and proficient in solving for c in infinite series.

The final answer is: $\boxed{\frac{1}{5}}$