Evaluate The Series: ∑ N = 1 ∞ 8 ⋅ ( 5 2 ) N \sum_{n=1}^{\infty} 8 \cdot\left(\frac{5}{2}\right)^n ∑ N = 1 ∞ ​ 8 ⋅ ( 2 5 ​ ) N

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Introduction

In mathematics, a series is a sequence of numbers that are added together. The series we will be evaluating in this article is $\sum_{n=1}^{\infty} 8 \cdot\left(\frac{5}{2}\right)^n$. This is a geometric series, which is a type of series where each term is obtained by multiplying the previous term by a fixed constant. In this case, the fixed constant is $\frac{5}{2}$.

Understanding Geometric Series

A geometric series is a series of the form $\sum_{n=1}^{\infty} ar^n$, where $a$ is the first term and $r$ is the common ratio. The series we are evaluating is a geometric series with $a = 8 \cdot \left(\frac{5}{2}\right)$ and $r = \frac{5}{2}$. The sum of a geometric series can be calculated using the formula $\frac{a}{1-r}$, but only if $|r| < 1$. If $|r| \geq 1$, the series diverges.

Evaluating the Series

To evaluate the series, we need to check if $|r| < 1$. In this case, $r = \frac{5}{2}$, which is greater than 1. Therefore, the series diverges.

Proof of Divergence

To prove that the series diverges, we can use the following argument. Let $S_n$ be the sum of the first $n$ terms of the series. Then, we have:

$$S_n = 8 \cdot \left(\frac{5}{2}\right) + 8 \cdot \left(\frac{5}{2}\right)^2 + \cdots + 8 \cdot \left(\frac{5}{2}\right)^n$$

We can factor out the common term $8 \cdot \left(\frac{5}{2}\right)$ from each term:

$$S_n = 8 \cdot \left(\frac{5}{2}\right) \cdot \left(1 + \left(\frac{5}{2}\right) + \left(\frac{5}{2}\right)^2 + \cdots + \left(\frac{5}{2}\right)^{n-1}\right)$$

The expression inside the parentheses is a geometric series with $a = 1$ and $r = \frac{5}{2}$. Since $|r| > 1$, this series diverges. Therefore, $S_n$ also diverges.

Conclusion

In conclusion, the series $\sum_{n=1}^{\infty} 8 \cdot\left(\frac{5}{2}\right)^n$ diverges because $|r| > 1$. This means that the sum of the series does not converge to a finite value.

Applications of Geometric Series

Geometric series have many applications in mathematics and other fields. Some examples include:

• Finance: Geometric series are used to calculate the future value of an investment.
• Physics: Geometric series are used to calculate the energy of a particle in a potential well.
• Computer Science: Geometric series are used to calculate the time complexity of algorithms.

Final Thoughts

In this article, we evaluated the series $\sum_{n=1}^{\infty} 8 \cdot\left(\frac{5}{2}\right)^n$ and showed that it diverges. We also discussed the applications of geometric series in various fields. Geometric series are a fundamental concept in mathematics, and understanding them is essential for many applications.

References

• Wikipedia: Geometric series
• Khan Academy: Geometric series
• MIT OpenCourseWare: Geometric series

Further Reading

• Calculus: A First Course by Michael Spivak
• Mathematics for Computer Science by Eric Lehman and Tom Leighton
• Geometric Series by Wikipedia

Related Topics

• Arithmetic Series: A series of the form $\sum_{n=1}^{\infty} a_n$, where $a_n$ is an arithmetic sequence.
• Harmonic Series: A series of the form $\sum_{n=1}^{\infty} \frac{1}{n}$.
• Convergence Tests: A set of tests used to determine whether a series converges or diverges.
  

Introduction

In our previous article, we evaluated the series $\sum_{n=1}^{\infty} 8 \cdot\left(\frac{5}{2}\right)^n$ and showed that it diverges. In this article, we will answer some frequently asked questions about geometric series.

Q: What is a geometric series?

A: A geometric series is a series of the form $\sum_{n=1}^{\infty} ar^n$, where $a$ is the first term and $r$ is the common ratio.

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

A: The formula for the sum of a geometric series is $\frac{a}{1-r}$, but only if $|r| < 1$. If $|r| \geq 1$, the series diverges.

Q: How do I determine if a geometric series converges or diverges?

A: To determine if a geometric series converges or diverges, you need to check if $|r| < 1$. If $|r| < 1$, the series converges. If $|r| \geq 1$, the series diverges.

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

A: A geometric series is a series of the form $\sum_{n=1}^{\infty} ar^n$, where $a$ is the first term and $r$ is the common ratio. An arithmetic series is a series of the form $\sum_{n=1}^{\infty} a_n$, where $a_n$ is an arithmetic sequence.

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 diverge if $|r| \geq 1$.

Q: Can a geometric series have a common ratio of 1?

A: Yes, a geometric series can have a common ratio of 1. However, the series will diverge if $a \neq 0$.

Q: What is the sum of the geometric series $\sum_{n=1}^{\infty} 2 \cdot 3^n$?

A: To find the sum of the geometric series $\sum_{n=1}^{\infty} 2 \cdot 3^n$, we need to check if $|r| < 1$. In this case, $r = 3$, which is greater than 1. Therefore, the series diverges.

Q: What is the sum of the geometric series $\sum_{n=1}^{\infty} 4 \cdot \left(\frac{1}{2}\right)^n$?

A: To find the sum of the geometric series $\sum_{n=1}^{\infty} 4 \cdot \left(\frac{1}{2}\right)^n$, we need to check if $|r| < 1$. In this case, $r = \frac{1}{2}$, which is less than 1. Therefore, the series converges, and the sum is $\frac{4}{1-\frac{1}{2}} = 8$.

Q: Can a geometric series have a common ratio of 0?

A: No, a geometric series cannot have a common ratio of 0. If $r = 0$, the series is a finite series, and its sum can be calculated using the formula for the sum of a finite series.

Q: What is the sum of the geometric series $\sum_{n=1}^{\infty} 6 \cdot 0^n$?

A: To find the sum of the geometric series $\sum_{n=1}^{\infty} 6 \cdot 0^n$, we need to check if $|r| < 1$. In this case, $r = 0$, which is less than 1. However, the series is a finite series, and its sum can be calculated using the formula for the sum of a finite series. In this case, the sum is 0.

Conclusion

In this article, we answered some frequently asked questions about geometric series. We discussed the formula for the sum of a geometric series, how to determine if a geometric series converges or diverges, and some examples of geometric series. We also discussed some common misconceptions about geometric series and provided some examples of geometric series with negative common ratios and common ratios of 1.

References

• Wikipedia: Geometric series
• Khan Academy: Geometric series
• MIT OpenCourseWare: Geometric series

Further Reading

• Calculus: A First Course by Michael Spivak
• Mathematics for Computer Science by Eric Lehman and Tom Leighton
• Geometric Series by Wikipedia

Related Topics

• Arithmetic Series: A series of the form $\sum_{n=1}^{\infty} a_n$, where $a_n$ is an arithmetic sequence.
• Harmonic Series: A series of the form $\sum_{n=1}^{\infty} \frac{1}{n}$.
• Convergence Tests: A set of tests used to determine whether a series converges or diverges.