Geometric Series Convergence

Oct 13, 2025 by magicalceylontours.com 5 views

Introduction

A geometric series is a type of infinite series where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. In this article, we will explore the convergence of a given geometric series and determine the value of a specific term in the series.

Geometric Series Convergence

A geometric series is convergent if the absolute value of the common ratio is less than 1. If the absolute value of the common ratio is greater than or equal to 1, the series is divergent.

Given the geometric series $\sum_{k=1}^{\infty} 4.3^{2-k}$ , we need to determine whether it is convergent or divergent.

The common ratio of the series is $r = \frac{1}{4.3}$ . Since $|r| = \frac{1}{4.3} < 1$ , the series is convergent.

Justification

To justify the convergence of the 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.

In this case, the first term of the series is $a = 4.3^1 = 4.3$ .

Substituting the values of $a$ and $r$ into the formula, we get:

S = \frac{4.3}{1 - \frac{1}{4.3}}

Simplifying the expression, we get:

S = \frac{4.3}{\frac{4.3 - 1}{4.3}}

S = \frac{4.3}{\frac{3.3}{4.3}}

S = \frac{4.3 \times 4.3}{3.3}

S = \frac{18.49}{3.3}

S = 5.61

Since the sum of the series is finite, the series is convergent.

Value Determination

Given that $\sum_{k=p}^{\infty} 4.3^{2-k}=\frac{2}{9}$ , we need to determine the value of $p$ .

To do this, 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.

In this case, the first term of the series is $a = 4.3^{2-p}$ .

Substituting the values of $a$ and $r$ into the formula, we get:

\frac{2}{9} = \frac{4.3^{2-p}}{1 - \frac{1}{4.3}}

Simplifying the expression, we get:

\frac{2}{9} = \frac{4.3^{2-p}}{\frac{4.3 - 1}{4.3}}

\frac{2}{9} = \frac{4.3^{2-p} \times 4.3}{3.3}

\frac{2}{9} = \frac{4.3^{2-p+1}}{3.3}

\frac{2}{9} = \frac{4.3^{3-p}}{3.3}

Cross-multiplying, we get:

2 \times 3.3 = 9 \times 4.3^{3-p}

6.6 = 39.27 \times 4.3^{3-p}

Dividing both sides by 39.27, we get:

\frac{6.6}{39.27} = 4.3^{3-p}

0.168 = 4.3^{3-p}

Taking the logarithm of both sides, we get:

\log(0.168) = \log(4.3^{3-p})

\log(0.168) = (3-p) \times \log(4.3)

Solving for $p$ , we get:

p = 3 - \frac{\log(0.168)}{\log(4.3)}

p = 3 - \frac{-0.73}{0.63}

p = 3 + 1.16

p = 4.16

Since $p$ must be an integer, we round down to the nearest integer:

p = 4

Therefore, the value of $p$ is 4.

Conclusion

In this article, we have determined that the geometric series $\sum_{k=1}^{\infty} 4.3^{2-k}$ is convergent. We have also determined the value of $p$ in the series $\sum_{k=p}^{\infty} 4.3^{2-k}=\frac{2}{9}$ to be 4.

References

[1] "Geometric Series" by Math Open Reference
[2] "Infinite Geometric Series" by Khan Academy
Geometric Series Q&A =========================

Q: What is a geometric series?

A: A geometric series is a type of infinite series where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio.

Q: What are the conditions for a geometric series to be convergent?

A: A geometric series is convergent if the absolute value of the common ratio is less than 1. If the absolute value of the common ratio is greater than or equal to 1, the series is divergent.

Q: How do you determine the sum of a convergent geometric series?

A: The sum of a convergent geometric series can be determined using the formula:

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

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

Q: What is the formula for the nth term of a geometric series?

A: The formula for the nth term of a geometric series is:

a_n = a \times r^{n-1}

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

Q: How do you determine the value of a specific term in a geometric series?

A: To determine the value of a specific term in a geometric series, you can use the formula:

a_n = a \times r^{n-1}

Substitute the values of $a$ , $r$ , and $n$ into the formula to find the value of the term.

Q: What is the relationship between the sum of a geometric series and the value of the common ratio?

A: The sum of a geometric series is inversely proportional to the value of the common ratio. As the common ratio approaches 1, the sum of the series approaches infinity.

Q: Can a geometric series have a negative common ratio?

A: Yes, a geometric series can have a negative common ratio. However, the series will still be convergent if the absolute value of the common ratio is less than 1.

Q: How do you determine the value of a specific term in a geometric series with a negative common ratio?

A: To determine the value of a specific term in a geometric series with a negative common ratio, you can use the formula:

a_n = a \times r^{n-1}

Substitute the values of $a$ , $r$ , and $n$ into the formula to find the value of the term.

Q: What is the relationship between the sum of a geometric series and the value of the first term?

A: The sum of a geometric series is directly proportional to the value of the first term. As the first term increases, the sum of the series also increases.

Q: Can a geometric series have a zero first term?

A: Yes, a geometric series can have a zero first term. However, the series will still be convergent if the absolute value of the common ratio is less than 1.

Q: How do you determine the value of a specific term in a geometric series with a zero first term?

A: To determine the value of a specific term in a geometric series with a zero first term, you can use the formula:

a_n = a \times r^{n-1}

Substitute the values of $a$ , $r$ , and $n$ into the formula to find the value of the term.

Q: What is the relationship between the sum of a geometric series and the number of terms?

A: The sum of a geometric series is inversely proportional to the number of terms. As the number of terms increases, the sum of the series approaches infinity.

Q: Can a geometric series have a fractional number of terms?

A: No, a geometric series cannot have a fractional number of terms. The number of terms must be a positive integer.

Q: How do you determine the value of a specific term in a geometric series with a fractional number of terms?

A: It is not possible to determine the value of a specific term in a geometric series with a fractional number of terms. The number of terms must be a positive integer.

Q: What is the relationship between the sum of a geometric series and the value of the common ratio squared?

A: The sum of a geometric series is inversely proportional to the value of the common ratio squared. As the common ratio squared approaches 1, the sum of the series approaches infinity.

Q: Can a geometric series have a common ratio squared equal to 1?

A: No, a geometric series cannot have a common ratio squared equal to 1. The common ratio must be less than 1 for the series to be convergent.

Q: How do you determine the value of a specific term in a geometric series with a common ratio squared equal to 1?

A: It is not possible to determine the value of a specific term in a geometric series with a common ratio squared equal to 1. The common ratio must be less than 1 for the series to be convergent.

Q: What is the relationship between the sum of a geometric series and the value of the first term squared?

A: The sum of a geometric series is directly proportional to the value of the first term squared. As the first term squared increases, the sum of the series also increases.

Q: Can a geometric series have a first term squared equal to 0?

A: Yes, a geometric series can have a first term squared equal to 0. However, the series will still be convergent if the absolute value of the common ratio is less than 1.

Q: How do you determine the value of a specific term in a geometric series with a first term squared equal to 0?

A: To determine the value of a specific term in a geometric series with a first term squared equal to 0, you can use the formula:

a_n = a \times r^{n-1}

Substitute the values of $a$ , $r$ , and $n$ into the formula to find the value of the term.

Q: What is the relationship between the sum of a geometric series and the value of the common ratio cubed?

A: The sum of a geometric series is inversely proportional to the value of the common ratio cubed. As the common ratio cubed approaches 1, the sum of the series approaches infinity.

Q: Can a geometric series have a common ratio cubed equal to 1?

A: No, a geometric series cannot have a common ratio cubed equal to 1. The common ratio must be less than 1 for the series to be convergent.

Q: How do you determine the value of a specific term in a geometric series with a common ratio cubed equal to 1?

A: It is not possible to determine the value of a specific term in a geometric series with a common ratio cubed equal to 1. The common ratio must be less than 1 for the series to be convergent.

Q: What is the relationship between the sum of a geometric series and the value of the first term cubed?

A: The sum of a geometric series is directly proportional to the value of the first term cubed. As the first term cubed increases, the sum of the series also increases.

Q: Can a geometric series have a first term cubed equal to 0?

A: Yes, a geometric series can have a first term cubed equal to 0. However, the series will still be convergent if the absolute value of the common ratio is less than 1.

Q: How do you determine the value of a specific term in a geometric series with a first term cubed equal to 0?

A: To determine the value of a specific term in a geometric series with a first term cubed equal to 0, you can use the formula:

a_n = a \times r^{n-1}

Substitute the values of $a$ , $r$ , and $n$ into the formula to find the value of the term.