Understanding Alternating Series

Oct 13, 2025 by magicalceylontours.com 62 views

Introduction

Alternating series are a type of infinite series where the terms alternate between positive and negative. In this problem, we are given an alternating series with the general term $a_n = (-1)^{n+1} \frac{(0.4)^{2n-1}}{(2n-1)!}$ . The series is given by $\sum_{n=1}^{\infty} a_n=0.4-\frac{(0.4)^3}{3!}+\frac{(0.4)^5}{5!}-\frac{(0.4)^7}{7!}+\ldots$ . We are asked to determine how many terms we need to compute in order for our partial sum to be within a certain tolerance of the infinite sum.

Understanding Alternating Series

Alternating series have the general form $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$ , where $a_n$ is a positive term for all $n$ . The terms of an alternating series alternate between positive and negative, and the absolute value of each term decreases as $n$ increases. This is known as the Alternating Series Test, which states that if the terms of an alternating series decrease in absolute value and approach zero, then the series converges.

Partial Sums and Tolerance

A partial sum of an infinite series is the sum of the first $n$ terms of the series. In this problem, we are interested in finding the number of terms we need to compute in order for our partial sum to be within a certain tolerance of the infinite sum. The tolerance is a measure of how close we want our partial sum to be to the infinite sum.

Computing Partial Sums

To compute the partial sum of an alternating series, we can use the following formula:

S_n = \sum_{k=1}^{n} (-1)^{k+1} a_k

where $S_n$ is the partial sum of the first $n$ terms, and $a_k$ is the $k$ th term of the series.

Determining the Number of Terms

To determine the number of terms we need to compute in order for our partial sum to be within a certain tolerance of the infinite sum, we can use the following inequality:

|S_n - S| \leq \frac{a_{n+1}}{1 - |a_{n+1}|}

where $S$ is the infinite sum, and $a_{n+1}$ is the $(n+1)$ th term of the series.

Applying the Inequality

We can apply the inequality to our problem by substituting the values of $a_n$ and $S$ . We have:

a_n = (-1)^{n+1} \frac{(0.4)^{2n-1}}{(2n-1)!}

and

S = \sum_{n=1}^{\infty} a_n = 0.4 - \frac{(0.4)^3}{3!} + \frac{(0.4)^5}{5!} - \frac{(0.4)^7}{7!} + \ldots

We want to find the number of terms we need to compute in order for our partial sum to be within a tolerance of $0.01$ of the infinite sum.

Computing the Number of Terms

We can compute the number of terms we need to compute by plugging in the values of $a_n$ and $S$ into the inequality and solving for $n$ . We get:

|S_n - S| \leq \frac{a_{n+1}}{1 - |a_{n+1}|}

\frac{a_{n+1}}{1 - |a_{n+1}|} \leq 0.01

a_{n+1} \leq 0.01(1 - |a_{n+1}|)

a_{n+1} \leq 0.01 - 0.01|a_{n+1}|

a_{n+1} + 0.01|a_{n+1}| \leq 0.01

a_{n+1}(1 + 0.01) \leq 0.01

a_{n+1} \leq \frac{0.01}{1.01}

a_{n+1} \leq 0.0099

We can now substitute the value of $a_n$ into the inequality and solve for $n$ . We get:

(-1)^{n+2} \frac{(0.4)^{2n+1}}{(2n+1)!} \leq 0.0099

\frac{(0.4)^{2n+1}}{(2n+1)!} \leq 0.0099

(0.4)^{2n+1} \leq 0.0099(2n+1)!

We can now take the logarithm of both sides of the inequality and solve for $n$ . We get:

\log(0.4)^{2n+1} \leq \log(0.0099(2n+1)!

(2n+1)\log(0.4) \leq \log(0.0099) + \log((2n+1)!

We can now use a calculator to solve for $n$ . We get:

n \leq 5

Therefore, we need to compute at least 5 terms in order for our partial sum to be within a tolerance of $0.01$ of the infinite sum.

Conclusion

In this problem, we were asked to determine how many terms we need to compute in order for our partial sum to be within a certain tolerance of the infinite sum. We used the Alternating Series Test to show that the series converges, and we used the inequality to determine the number of terms we need to compute. We found that we need to compute at least 5 terms in order for our partial sum to be within a tolerance of $0.01$ of the infinite sum.

References

[1] Apostol, T. M. (1974). Calculus. Vol. 1. New York: Wiley.
[2] Rudin, W. (1976). Principles of mathematical analysis. New York: McGraw-Hill.
[3] Spivak, M. (1965). Calculus. New York: W.A. Benjamin.

Note

Q: What is an alternating series?

A: An alternating series is a type of infinite series where the terms alternate between positive and negative. The general form of an alternating series is $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$ , where $a_n$ is a positive term for all $n$ .

Q: What is the Alternating Series Test?

A: The Alternating Series Test is a test used to determine whether an alternating series converges or diverges. The test states that if the terms of an alternating series decrease in absolute value and approach zero, then the series converges.

Q: How do I determine the number of terms I need to compute in order for my partial sum to be within a certain tolerance of the infinite sum?

A: To determine the number of terms you need to compute, you can use the inequality:

|S_n - S| \leq \frac{a_{n+1}}{1 - |a_{n+1}|}

where $S_n$ is the partial sum of the first $n$ terms, $S$ is the infinite sum, and $a_{n+1}$ is the $(n+1)$ th term of the series.

Q: What is the significance of the Alternating Series Test?

A: The Alternating Series Test is significant because it provides a way to determine whether an alternating series converges or diverges. This is important because it allows us to determine whether a series can be approximated by a finite sum.

Q: Can you give an example of an alternating series?

A: Yes, the series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{(0.4)^{2n-1}}{(2n-1)!}$ is an example of an alternating series.

Q: How do I compute the partial sum of an alternating series?

A: To compute the partial sum of an alternating series, you can use the formula:

S_n = \sum_{k=1}^{n} (-1)^{k+1} a_k

where $S_n$ is the partial sum of the first $n$ terms, and $a_k$ is the $k$ th term of the series.

Q: What is the difference between a partial sum and an infinite sum?

A: A partial sum is the sum of the first $n$ terms of a series, while an infinite sum is the sum of all the terms of a series.

Q: Can you explain the concept of tolerance in the context of alternating series?

A: Yes, tolerance refers to the amount of error that is allowed in the approximation of the infinite sum by a partial sum. In other words, it is the maximum amount by which the partial sum can differ from the infinite sum.

Q: How do I determine the tolerance for an alternating series?

A: To determine the tolerance for an alternating series, you can use the inequality:

|S_n - S| \leq \frac{a_{n+1}}{1 - |a_{n+1}|}

where $S_n$ is the partial sum of the first $n$ terms, $S$ is the infinite sum, and $a_{n+1}$ is the $(n+1)$ th term of the series.

Q: Can you give an example of how to use the inequality to determine the tolerance for an alternating series?

A: Yes, suppose we want to determine the tolerance for the alternating series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{(0.4)^{2n-1}}{(2n-1)!}$ . We can use the inequality:

|S_n - S| \leq \frac{a_{n+1}}{1 - |a_{n+1}|}

where $S_n$ is the partial sum of the first $n$ terms, $S$ is the infinite sum, and $a_{n+1}$ is the $(n+1)$ th term of the series.

We can now substitute the values of $a_n$ and $S$ into the inequality and solve for $n$ . We get:

n \leq 5

Therefore, we need to compute at least 5 terms in order for our partial sum to be within a tolerance of $0.01$ of the infinite sum.

Conclusion

In this article, we have discussed the Alternating Series Test and how to use it to determine whether an alternating series converges or diverges. We have also discussed how to use the inequality to determine the number of terms we need to compute in order for our partial sum to be within a certain tolerance of the infinite sum. We have given examples of how to use the inequality and have explained the concept of tolerance in the context of alternating series.