Problem 39 Evaluate each infinite series th... [FREE SOLUTION]

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Chapter 11: Problem 39

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty} 2(1.2)^{n-1} $$

Short Answer

Expert verified

The sum does not exist because the common ratio is greater than 1.

Step by step solution

Identify the first term and the common ratio

In the series $\sum_{n=1}^{\infty} 2(1.2)^{n-1}$, $a = 2$ and $r = 1.2$.

Check if the sum of the series exists

The sum of the series exists if the absolute value of the common ratio $|r|$ is less than 1. In this case, $|r| = 1.2$, which is greater than 1, so the sum doesn't exist.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Series

In mathematics, a geometric series is a series of numbers with a constant ratio between successive terms. This means that to go from one term to the next, you multiply by the same number each time. For example, in a geometric series like $2, 4, 8, 16, \ldots$, the common ratio $r$ is 2 because each term is 2 times the previous term.

A geometric series can be written in the form:

First term $a$
Common ratio $r$
Terms: $a, ar, ar^2, ar^3, \ldots$

The series continues indefinitely, adding up all these terms. It's crucial to notice whether a geometric series converges, which is an important concept when determining the sum.

Convergence

Convergence is a concept that indicates whether a series approaches a specific value as more of its terms are added. For a geometric series, convergence is determined by the value of the common ratio $r$.

If the common ratio's absolute value $|r|$ is less than 1, the series converges, meaning it will approach a finite sum as more and more terms are added. If $|r|$ is greater than or equal to 1, the series diverges, suggesting it does not settle to a finite value.

Convergent Series: $|r| < 1$
Divergent Series: $|r| \geq 1$

In our original exercise, $|r| = 1.2$ which is greater than 1, making the series divergent. This means it doesn't have a set sum and continues to grow indefinitely.

Sum of Series

The sum of a series is the value that the series approaches as an infinite number of terms are added together. For a convergent geometric series, there is a convenient formula to find the sum:

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

where $a$ is the first term and $r$ is the common ratio. It's important to remember that this formula only works if the series is convergent $(|r| < 1)$.

In cases where $|r|$ is not less than 1, like our original problem, the series does not have a sum because it diverges. The terms keep increasing and do not approach a fixed value. This is why determining the convergence before finding the sum is so crucial in geometric series.

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91影视

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty} 2(1.2)^{n-1} $$

Short Answer

Step by step solution

Identify the first term and the common ratio

Check if the sum of the series exists

Key Concepts

Geometric Series

Convergence

Sum of Series

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