How Many Terms Of The Series Do We Need To Add In Order To Find The Sum To The Indicated Accuracy? Your Answer Must Be The Smallest Possible Integer.$\sum_{n=1} {\infty}(-1) {n-1} \frac{6}{n^4}, \quad \mid \text{error} \mid \ \textless \

Introduction

In mathematics, the concept of series and their convergence is a crucial topic in understanding various mathematical concepts. A series is the sum of the terms of a sequence. The convergence of a series is determined by the behavior of its terms as the index of the term increases. In this article, we will discuss how to find the number of terms of a series that need to be added in order to find the sum to the indicated accuracy.

Understanding the Problem

The problem at hand is to find the number of terms of the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{6}{n^4}$ that need to be added in order to find the sum to the indicated accuracy. The series is an alternating series, meaning that the terms alternate between positive and negative. The general term of the series is given by $(-1)^{n-1} \frac{6}{n^4}$.

The Alternating Series Test

The alternating series test is a test used to determine the convergence of an alternating series. The test states that if the terms of the series are alternately positive and negative, and the absolute value of the terms decreases monotonically to zero, then the series converges. In this case, the terms of the series are alternately positive and negative, and the absolute value of the terms decreases monotonically to zero.

The Error Bound

The error bound is a measure of the accuracy of the partial sum of a series. It is defined as the absolute value of the difference between the partial sum and the actual sum of the series. In this case, we want to find the number of terms that need to be added in order to find the sum to the indicated accuracy, which means we want to find the number of terms that need to be added in order to make the error bound less than a given value.

The Remainder

The remainder is the difference between the actual sum of the series and the partial sum. It is also known as the error. In this case, we want to find the number of terms that need to be added in order to make the remainder less than a given value.

The Integral Test

The integral test is a test used to determine the convergence of a series. It states that if the function $f(x)$ is positive, continuous, and decreasing on the interval $[1, \infty)$, and if the integral $\int_{1}^{\infty} f(x) dx$ converges, then the series $\sum_{n=1}^{\infty} f(n)$ converges. In this case, we can use the integral test to determine the convergence of the series.

The Integral Test for the Given Series

To apply the integral test, we need to find the function $f(x)$ such that $f(n) = (-1)^{n-1} \frac{6}{n^4}$. We can see that $f(x) = (-1)^{x-1} \frac{6}{x^4}$. The function $f(x)$ is positive, continuous, and decreasing on the interval $[1, \infty)$.

The Integral

To find the integral, we can use the following formula:

$$\int_{1}^{\infty} f(x) dx = \lim_{b \to \infty} \int_{1}^{b} f(x) dx$$

We can evaluate the integral as follows:

$$\int_{1}^{b} f(x) dx = \int_{1}^{b} (-1)^{x-1} \frac{6}{x^4} dx$$

$$= 6 \int_{1}^{b} \frac{(-1)^{x-1}}{x^4} dx$$

$$= 6 \left[ \frac{(-1)^{x-1}}{3x^3} \right]_{1}^{b}$$

$$= 2 \left[ \frac{(-1)^{x-1}}{x^3} \right]_{1}^{b}$$

The Limit

To find the limit, we can use the following formula:

$$\lim_{b \to \infty} \frac{(-1)^{b-1}}{b^3} = 0$$

We can see that the limit exists and is equal to zero.

The Convergence of the Series

Since the integral converges, we can conclude that the series converges.

The Error Bound

To find the error bound, we can use the following formula:

$$\mid \text{error} \mid \ \textless \ \frac{M}{n+1}$$

where $M$ is the maximum value of the terms of the series, and $n$ is the number of terms that have been added.

The Maximum Value

To find the maximum value of the terms of the series, we can use the following formula:

$$M = \max \left\{ \left| \frac{6}{n^4} \right| \right\}$$

We can see that the maximum value of the terms of the series is equal to $\frac{6}{1^4} = 6$.

The Error Bound Formula

We can substitute the value of $M$ into the error bound formula as follows:

$$\mid \text{error} \mid \ \textless \ \frac{6}{n+1}$$

The Number of Terms

To find the number of terms that need to be added in order to make the error bound less than a given value, we can use the following formula:

$$n = \left\lceil \frac{6}{\epsilon} - 1 \right\rceil$$

where $\epsilon$ is the given value of the error bound.

The Solution

To find the number of terms that need to be added in order to find the sum to the indicated accuracy, we can use the following formula:

$$n = \left\lceil \frac{6}{\epsilon} - 1 \right\rceil$$

We can substitute the value of $\epsilon$ into the formula as follows:

$$n = \left\lceil \frac{6}{0.01} - 1 \right\rceil$$

$$= \left\lceil 600 - 1 \right\rceil$$

$$= 599$$

Conclusion

In conclusion, we have found that the number of terms that need to be added in order to find the sum to the indicated accuracy is 599. This is the smallest possible integer.

References

• [1] "Calculus" by Michael Spivak
• [2] "Real and Complex Analysis" by Walter Rudin
• [3] "Introduction to Real Analysis" by Bartle and Sherbert

Glossary

• Alternating series: A series whose terms alternate between positive and negative.
• Convergence: The property of a series that its partial sums converge to a limit.
• Error bound: A measure of the accuracy of the partial sum of a series.
• Integral test: A test used to determine the convergence of a series.
• Maximum value: The largest value of a function.
• Partial sum: The sum of the first $n$ terms of a series.
• Remainder: The difference between the actual sum of a series and the partial sum.
• Series: The sum of the terms of a sequence.
  

Introduction

In our previous article, we discussed how to find the number of terms of a series that need to be added in order to find the sum to the indicated accuracy. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the Alternating Series Test?

A: The Alternating Series Test is a test used to determine the convergence of an alternating series. The test states that if the terms of the series are alternately positive and negative, and the absolute value of the terms decreases monotonically to zero, then the series converges.

Q: What is the Error Bound?

A: The error bound is a measure of the accuracy of the partial sum of a series. It is defined as the absolute value of the difference between the partial sum and the actual sum of the series.

Q: How Do We Find the Error Bound?

A: To find the error bound, we can use the following formula:

$$\mid \text{error} \mid \ \textless \ \frac{M}{n+1}$$

where $M$ is the maximum value of the terms of the series, and $n$ is the number of terms that have been added.

Q: What is the Maximum Value?

A: The maximum value is the largest value of a function. In the context of the Alternating Series Test, the maximum value is the largest value of the terms of the series.

Q: How Do We Find the Maximum Value?

A: To find the maximum value, we can use the following formula:

$$M = \max \left\{ \left| \frac{6}{n^4} \right| \right\}$$

Q: What is the Integral Test?

A: The Integral Test is a test used to determine the convergence of a series. It states that if the function $f(x)$ is positive, continuous, and decreasing on the interval $[1, \infty)$, and if the integral $\int_{1}^{\infty} f(x) dx$ converges, then the series $\sum_{n=1}^{\infty} f(n)$ converges.

Q: How Do We Apply the Integral Test?

A: To apply the Integral Test, we need to find the function $f(x)$ such that $f(n) = (-1)^{n-1} \frac{6}{n^4}$. We can see that $f(x) = (-1)^{x-1} \frac{6}{x^4}$. The function $f(x)$ is positive, continuous, and decreasing on the interval $[1, \infty)$.

Q: What is the Remainder?

A: The remainder is the difference between the actual sum of a series and the partial sum. It is also known as the error.

Q: How Do We Find the Remainder?

A: To find the remainder, we can use the following formula:

$$\mid \text{remainder} \mid \ \textless \ \frac{M}{n+1}$$

Q: What is the Partial Sum?

A: The partial sum is the sum of the first $n$ terms of a series.

Q: How Do We Find the Partial Sum?

A: To find the partial sum, we can use the following formula:

$$S_n = \sum_{k=1}^{n} a_k$$

where $a_k$ is the $k$th term of the series.

Q: What is the Actual Sum?

A: The actual sum is the sum of the infinite series.

Q: How Do We Find the Actual Sum?

A: To find the actual sum, we can use the following formula:

$$S = \lim_{n \to \infty} S_n$$

Q: What is the Convergence of a Series?

A: The convergence of a series is the property of a series that its partial sums converge to a limit.

Q: How Do We Determine the Convergence of a Series?

A: To determine the convergence of a series, we can use the Alternating Series Test, the Integral Test, or other tests.

Q: What is the Accuracy of a Series?

A: The accuracy of a series is the measure of how close the partial sum is to the actual sum.

Q: How Do We Determine the Accuracy of a Series?

A: To determine the accuracy of a series, we can use the error bound formula.

Conclusion

In conclusion, we have answered some of the most frequently asked questions related to the topic of how many terms of a series need to be added in order to find the sum to the indicated accuracy. We hope that this article has been helpful in understanding this topic.

References

• [1] "Calculus" by Michael Spivak
• [2] "Real and Complex Analysis" by Walter Rudin
• [3] "Introduction to Real Analysis" by Bartle and Sherbert

Glossary

• Alternating series: A series whose terms alternate between positive and negative.
• Convergence: The property of a series that its partial sums converge to a limit.
• Error bound: A measure of the accuracy of the partial sum of a series.
• Integral test: A test used to determine the convergence of a series.
• Maximum value: The largest value of a function.
• Partial sum: The sum of the first $n$ terms of a series.
• Remainder: The difference between the actual sum of a series and the partial sum.
• Series: The sum of the terms of a sequence.