Choosing the Correct Expansion

Oct 13, 2025 by magicalceylontours.com 20 views

In mathematics, a series is the sum of the terms of a sequence. The given problem involves finding the sum of a series with a specific pattern. The series is defined as $\sum_{n=1}$ , where $n$ ranges from 1 to 6.

Breaking Down the Series

To find the sum of the series, we need to expand it and then add up the terms. The series can be expanded as follows:

$\sum_{n=1}^6(-2)^{n-1} = (-2)^{0} + (-2)^{1} + (-2)^{2} + (-2)^{3} + (-2)^{4} + (-2)^{5}$

Evaluating the Terms

Now, let's evaluate each term in the series:

$(-2)^{0} = 1$
$(-2)^{1} = -2$
$(-2)^{2} = 4$
$(-2)^{3} = -8$
$(-2)^{4} = 16$
$(-2)^{5} = -32$

Choosing the Correct Expansion

Now that we have evaluated each term, let's compare the options given:

A. $-1 + 2 + (-4) + 8 + (-16) + 32$ B. $1 + (-2) + 4 + (-8) + 16 + (-32)$ C. $1 + 2 + 4 + 8 + 16 + (-32)$

Based on our evaluation, the correct expansion is:

B. $1 + (-2) + 4 + (-8) + 16 + (-32)$

Why is this the Correct Expansion?

The correct expansion is option B because it matches the evaluated terms we obtained earlier. The series is defined as $\sum_{n=1}^6(-2)^{n-1}$ , and when we expand it, we get the terms $1, -2, 4, -8, 16, -32$ . This matches option B, which is the correct expansion of the series.

Conclusion

In the previous article, we discussed the expansion of the series $\sum_{n=1}^6(-2)^{n-1}$ . In this article, we will answer some frequently asked questions (FAQs) about series expansion.

Q: What is a series expansion?

A: A series expansion is the process of expressing a function or a mathematical expression as a sum of terms. In the case of the series $\sum_{n=1}^6(-2)^{n-1}$ , we expanded it to get the terms $1, -2, 4, -8, 16, -32$ .

Q: How do I know which terms to include in the series expansion?

A: To determine which terms to include in the series expansion, you need to identify the pattern of the series. In the case of the series $\sum_{n=1}^6(-2)^{n-1}$ , the pattern is obtained by raising $-2$ to the power of $n-1$ , where $n$ ranges from 1 to 6.

Q: What is the difference between a series and a sequence?

A: A sequence is a list of numbers or terms that are arranged in a specific order. A series, on the other hand, is the sum of the terms of a sequence. In other words, a series is a sequence of sums.

Q: How do I evaluate the terms of a series expansion?

A: To evaluate the terms of a series expansion, you need to substitute the values of $n$ into the expression and simplify. In the case of the series $\sum_{n=1}^6(-2)^{n-1}$ , we substituted the values of $n$ into the expression $(-2)^{n-1}$ and simplified to get the terms $1, -2, 4, -8, 16, -32$ .

Q: What is the significance of the power of $-2$ in the series expansion?

A: The power of $-2$ in the series expansion determines the sign of each term. In the case of the series $\sum_{n=1}^6(-2)^{n-1}$ , the power of $-2$ is $n-1$ , which means that the sign of each term alternates between positive and negative.

Q: Can I use a calculator to evaluate the terms of a series expansion?

A: Yes, you can use a calculator to evaluate the terms of a series expansion. However, it's always a good idea to double-check your work by hand to ensure that you get the correct answer.

Q: How do I know if a series expansion is convergent or divergent?

A: A series expansion is convergent if the sum of the terms approaches a finite value as the number of terms increases. A series expansion is divergent if the sum of the terms approaches infinity as the number of terms increases.

Conclusion

In conclusion, series expansion is an important concept in mathematics that involves expressing a function or a mathematical expression as a sum of terms. By understanding the basics of series expansion, you can evaluate the terms of a series and determine whether it is convergent or divergent.

Common Mistakes to Avoid

When working with series expansion, it's easy to make mistakes. Here are some common mistakes to avoid:

Not identifying the pattern of the series: Make sure to identify the pattern of the series before expanding it.
Not evaluating the terms correctly: Double-check your work by hand to ensure that you get the correct answer.
Not considering the power of the term: The power of the term determines the sign of each term, so make sure to consider it when evaluating the terms.
Not checking for convergence: Make sure to check whether the series expansion is convergent or divergent before using it.

Additional Resources

If you're interested in learning more about series expansion, here are some additional resources:

Math textbooks: Check out math textbooks that cover series expansion, such as "Calculus" by Michael Spivak or "Mathematics for the Nonmathematician" by Morris Kline.
Online resources: Check out online resources such as Khan Academy, MIT OpenCourseWare, or Wolfram Alpha.
Practice problems: Practice solving series expansion problems to get a feel for how to evaluate the terms and determine whether the series expansion is convergent or divergent.