Which Equation Represents The Partial Sum Of The Geometric Series?A. $\sum_{n=1}^3\left(\frac{1}{2}\right) \cdot\left(\frac{1}{2}\right)^{n-1}$B. $\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$C.

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The partial sum of a geometric series is the sum of the first n terms of the series. In this article, we will explore the concept of partial sums of geometric series and determine which equation represents the partial sum of a given geometric series.

What is a Geometric Series?

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric series is:

$$a, ar, ar^2, ar^3, \ldots$$

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

Example of a Geometric Series

Consider the geometric series:

$$2, 2 \cdot \frac{1}{2}, 2 \cdot \left(\frac{1}{2}\right)^2, 2 \cdot \left(\frac{1}{2}\right)^3, \ldots$$

In this series, the first term is 2 and the common ratio is $\frac{1}{2}$.

Partial Sums of Geometric Series

The partial sum of a geometric series is the sum of the first n terms of the series. The formula for the partial sum of a geometric series is:

$$S_n = a \cdot \frac{1 - r^n}{1 - r}$$

where $S_n$ is the partial sum, a is the first term, r is the common ratio, and n is the number of terms.

Which Equation Represents the Partial Sum of the Geometric Series?

Let's examine the given equations and determine which one represents the partial sum of the geometric series.

Equation A

$$\sum_{n=1}^3\left(\frac{1}{2}\right) \cdot\left(\frac{1}{2}\right)^{n-1}$$

This equation represents the sum of the first three terms of the geometric series with first term $\frac{1}{2}$ and common ratio $\frac{1}{2}$. The partial sum of the first three terms is:

$$\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(\frac{1}{2}\right)^2$$

Simplifying the expression, we get:

$$\frac{1}{2} + \frac{1}{4} + \frac{1}{8}$$

Equation B

$$\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

This equation represents the sum of the first three terms of the geometric series with first term $\frac{1}{4}$ and common ratio $\frac{1}{2}$. The partial sum of the first three terms is:

$$\frac{1}{4} + \frac{1}{8} + \frac{1}{16}$$

Equation C

There is no equation C provided.

Conclusion

Based on the analysis, Equation A represents the partial sum of the geometric series with first term $\frac{1}{2}$ and common ratio $\frac{1}{2}$. Equation B represents the sum of the first three terms of a different geometric series with first term $\frac{1}{4}$ and common ratio $\frac{1}{2}$.

Key Takeaways

• A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
• The partial sum of a geometric series is the sum of the first n terms of the series.
• The formula for the partial sum of a geometric series is $S_n = a \cdot \frac{1 - r^n}{1 - r}$.
• Equation A represents the partial sum of the geometric series with first term $\frac{1}{2}$ and common ratio $\frac{1}{2}$.

Further Reading

For more information on geometric series and partial sums, please refer to the following resources:

• Geometric Series
• Partial Sums of Geometric Series
• Mathematics

References

• [1] "Geometric Series" by Wikipedia
• [2] "Partial Sums of Geometric Series" by Wikipedia
• [3] "Mathematics" by Wikipedia
  Geometric Series and Partial Sums: Frequently Asked Questions =============================================================

In this article, we will address some of the most frequently asked questions about geometric series and partial sums.

Q: What is a geometric series?

A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

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

A: The formula for the partial sum of a geometric series is:

$$S_n = a \cdot \frac{1 - r^n}{1 - r}$$

where $S_n$ is the partial sum, a is the first term, r is the common ratio, and n is the number of terms.

Q: How do I determine the common ratio of a geometric series?

A: To determine the common ratio of a geometric series, you can divide any term by the previous term. For example, if the series is:

$$2, 4, 8, 16, \ldots$$

You can divide the second term (4) by the first term (2) to get the common ratio:

$$\frac{4}{2} = 2$$

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

A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. An arithmetic series, on the other hand, is a sequence of numbers in which each term after the first is found by adding a fixed number called the common difference.

Q: Can I use the formula for the partial sum of a geometric series if the common ratio is 1?

A: No, you cannot use the formula for the partial sum of a geometric series if the common ratio is 1. In this case, the series is not geometric, and you should use a different formula.

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

A: To find the sum of an infinite geometric series, you can use the formula:

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

where S is the sum, a is the first term, and r is the common ratio.

Q: What is the difference between a convergent and a divergent geometric series?

A: A convergent geometric series is a series that has a finite sum, while a divergent geometric series is a series that has an infinite sum.

Q: Can I use the formula for the partial sum of a geometric series if the common ratio is greater than 1?

A: Yes, you can use the formula for the partial sum of a geometric series if the common ratio is greater than 1. However, the series may be divergent, and the sum may not exist.

Q: How do I determine if a geometric series is convergent or divergent?

A: To determine if a geometric series is convergent or divergent, you can use the following criteria:

• If the absolute value of the common ratio is less than 1, the series is convergent.
• If the absolute value of the common ratio is greater than 1, the series is divergent.
• If the common ratio is equal to 1, the series is not geometric, and you should use a different formula.

Q: Can I use the formula for the partial sum of a geometric series if the series has a negative common ratio?

A: Yes, you can use the formula for the partial sum of a geometric series if the series has a negative common ratio. However, the series may be divergent, and the sum may not exist.

Q: How do I find the sum of a geometric series with a negative common ratio?

A: To find the sum of a geometric series with a negative common ratio, you can use the formula:

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

where S is the sum, a is the first term, and r is the common ratio.

Q: Can I use the formula for the partial sum of a geometric series if the series has a complex common ratio?

A: Yes, you can use the formula for the partial sum of a geometric series if the series has a complex common ratio. However, the series may be divergent, and the sum may not exist.

Q: How do I find the sum of a geometric series with a complex common ratio?

A: To find the sum of a geometric series with a complex common ratio, you can use the formula:

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

where S is the sum, a is the first term, and r is the common ratio.

Conclusion

In this article, we have addressed some of the most frequently asked questions about geometric series and partial sums. We hope that this information has been helpful in understanding these concepts. If you have any further questions, please do not hesitate to ask.