Determine Whether Each Of The Following Series Converges. For The Series That Converge, Enter The Sum.(a) $\sum_{n=1}^{\infty} \frac{8^n}{7^n} = $ $\square$(b) $\sum_{n=2}^{\infty} \frac{1}{3^n} = $ $\square$(c)

Feb 28, 2025 by ADMIN 212 views

**Convergence of Infinite Series: A Comprehensive Analysis**

Introduction

Infinite series are a fundamental concept in mathematics, and understanding their convergence is crucial in various fields, including calculus, analysis, and physics. In this article, we will delve into the world of infinite series and determine whether each of the given series converges. We will also calculate the sum of the convergent series.

Convergence Tests

To determine the convergence of an infinite series, we can use various convergence tests. Some of the most commonly used tests include:

Geometric Series Test: This test is used to determine the convergence of a geometric series, which is a series of the form $\sum_{n=1}^{\infty} ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.
Ratio Test: This test is used to determine the convergence of a series by comparing the ratio of consecutive terms.
Root Test: This test is used to determine the convergence of a series by comparing the $n$ th root of the $n$ th term.

Series (a)

The first series we will analyze is $\sum_{n=1}^{\infty} \frac{8^n}{7^n}$ . This is a geometric series with a common ratio of $\frac{8}{7}$ .

Geometric Series Test

To determine the convergence of this series, we can use the Geometric Series Test. The test states that a geometric series converges if and only if the absolute value of the common ratio is less than 1.

\left|\frac{8}{7}\right| < 1

Since the absolute value of the common ratio is less than 1, the series converges.

Calculating the Sum

To calculate the sum of the series, we can use the formula for the sum of a geometric series:

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

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

In this case, the first term is $\frac{8}{7}$ , and the common ratio is also $\frac{8}{7}$ . Plugging these values into the formula, we get:

S = \frac{\frac{8}{7}}{1 - \frac{8}{7}}

Simplifying the expression, we get:

S = \frac{\frac{8}{7}}{-\frac{1}{7}}

S = -8

Therefore, the sum of the series is $-8$ .

Series (b)

The second series we will analyze is $\sum_{n=2}^{\infty} \frac{1}{3^n}$ . This is also a geometric series with a common ratio of $\frac{1}{3}$ .

Geometric Series Test

\left|\frac{1}{3}\right| < 1

Since the absolute value of the common ratio is less than 1, the series converges.

Calculating the Sum

To calculate the sum of the series, we can use the formula for the sum of a geometric series:

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

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

In this case, the first term is $\frac{1}{3^2} = \frac{1}{9}$ , and the common ratio is $\frac{1}{3}$ . Plugging these values into the formula, we get:

S = \frac{\frac{1}{9}}{1 - \frac{1}{3}}

Simplifying the expression, we get:

S = \frac{\frac{1}{9}}{\frac{2}{3}}

S = \frac{1}{6}

Therefore, the sum of the series is $\frac{1}{6}$ .

Conclusion

In this article, we analyzed two infinite series and determined whether they converge. We also calculated the sum of the convergent series. The first series, $\sum_{n=1}^{\infty} \frac{8^n}{7^n}$ , converges with a sum of $-8$ . The second series, $\sum_{n=2}^{\infty} \frac{1}{3^n}$ , also converges with a sum of $\frac{1}{6}$ . These results demonstrate the importance of understanding the convergence of infinite series in mathematics and its applications.

References

[1] Kolmogorov, A. N. (1950). Foundations of the Theory of Probability. Chelsea Publishing Company.
[2] Walter Rudin (1976). Principles of Mathematical Analysis. McGraw-Hill Book Company.
[3] Spivak, M. (1965). Calculus. W.A. Benjamin, Inc.

Introduction

Infinite series are a fundamental concept in mathematics, and understanding their convergence is crucial in various fields, including calculus, analysis, and physics. In this article, we will answer some frequently asked questions about infinite series and provide additional insights into their properties and applications.

Q: What is an infinite series?

A: An infinite series is the sum of an infinite number of terms, where each term is a function of the index of the term. It is often denoted as $\sum_{n=1}^{\infty} a_n$ , where $a_n$ is the $n$ th term of the series.

Q: What are the different types of infinite series?

A: There are several types of infinite series, including:

Geometric series: A series of the form $\sum_{n=1}^{\infty} ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.
Arithmetic series: A series of the form $\sum_{n=1}^{\infty} a + (n-1)d$ , where $a$ is the first term and $d$ is the common difference.
Power series: A series of the form $\sum_{n=0}^{\infty} a_n x^n$ , where $a_n$ is the coefficient of the $n$ th term and $x$ is the variable.

Q: How do I determine if an infinite series converges?

A: There are several tests that can be used to determine if an infinite series converges, including:

Geometric Series Test: This test is used to determine the convergence of a geometric series, which is a series of the form $\sum_{n=1}^{\infty} ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.
Ratio Test: This test is used to determine the convergence of a series by comparing the ratio of consecutive terms.
Root Test: This test is used to determine the convergence of a series by comparing the $n$ th root of the $n$ th term.

Q: What is the sum of an infinite series?

A: The sum of an infinite series is the value that the series approaches as the number of terms increases without bound. It is often denoted as $S$ .

Q: How do I calculate the sum of an infinite series?

A: The sum of an infinite series can be calculated using various methods, including:

Geometric Series Formula: This formula is used to calculate the sum of a geometric series: $S = \frac{a}{1 - r}$ , where $a$ is the first term and $r$ is the common ratio.
Power Series Formula: This formula is used to calculate the sum of a power series: $S = \sum_{n=0}^{\infty} a_n x^n$ , where $a_n$ is the coefficient of the $n$ th term and $x$ is the variable.

Q: What are some common applications of infinite series?

A: Infinite series have numerous applications in various fields, including:

Calculus: Infinite series are used to represent functions as power series, which can be used to approximate the function.
Analysis: Infinite series are used to study the properties of functions, such as convergence and continuity.
Physics: Infinite series are used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.

Conclusion

In this article, we have answered some frequently asked questions about infinite series and provided additional insights into their properties and applications. We hope that this article has been helpful in understanding the concept of infinite series and their importance in mathematics and other fields.

References

[1] Kolmogorov, A. N. (1950). Foundations of the Theory of Probability. Chelsea Publishing Company.
[2] Walter Rudin (1976). Principles of Mathematical Analysis. McGraw-Hill Book Company.
[3] Spivak, M. (1965). Calculus. W.A. Benjamin, Inc.

Determine Whether Each Of The Following Series Converges. For The Series That Converge, Enter The Sum.(a) $\sum_{n=1}^{\infty} \frac{8^n}{7^n} = $ $\square$(b) $\sum_{n=2}^{\infty} \frac{1}{3^n} = $ $\square$(c)

Introduction

Convergence Tests

Series (a)

Geometric Series Test

Calculating the Sum

Series (b)

Geometric Series Test

Calculating the Sum

Conclusion

References

Further Reading

Introduction

Q: What is an infinite series?

Q: What are the different types of infinite series?

Q: How do I determine if an infinite series converges?

Q: What is the sum of an infinite series?

Q: How do I calculate the sum of an infinite series?

Q: What are some common applications of infinite series?

Conclusion

References

Further Reading