The Series and Its Convergence

Oct 13, 2025 by talynt.org 8 views

Introduction

Infinite series are a fundamental concept in mathematics, and approximating their values is a crucial task in various fields, including calculus, analysis, and numerical methods. In this article, we will focus on approximating the value of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^5}$ to within an error of at most $10^{-4}$ .

The Series and Its Convergence

The given series is an alternating series, which means that the terms alternate between positive and negative values. The series is defined as:

\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^5}

This series is a special case of the alternating series, where the terms are of the form $\frac{(-1)^{n+1}}{n^p}$ , with $p=5$ . The series converges according to the Alternating Series Test, which states that if the terms of the series are alternating and decreasing in absolute value, and if the limit of the terms is zero, then the series converges.

The Error Bound

To approximate the value of the series, we need to find the error bound, which is the maximum possible error between the partial sum and the actual value of the series. According to Equation (2), the error bound is given by:

\left|S_N-S\right| \leq a_{N+1}

where $S_N$ is the partial sum of the series up to the $N$ -th term, and $a_{N+1}$ is the $(N+1)$ -th term of the series.

Finding the Smallest Value of N

To find the smallest value of $N$ that satisfies the error bound, we need to find the value of $N$ such that:

a_{N+1} \leq 10^{-4}

Since the terms of the series are of the form $\frac{(-1)^{n+1}}{n^5}$ , we can write:

a_{N+1} = \frac{1}{(N+1)^5}

Substituting this expression into the error bound equation, we get:

\frac{1}{(N+1)^5} \leq 10^{-4}

Solving for $N$ , we get:

(N+1)^5 \geq 10^4

Taking the fifth root of both sides, we get:

N+1 \geq 10

Subtracting 1 from both sides, we get:

N \geq 9

Therefore, the smallest value of $N$ that satisfies the error bound is $N=9$ .

Conclusion

In this article, we have approximated the value of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^5}$ to within an error of at most $10^{-4}$ . We have used the Alternating Series Test to establish the convergence of the series and the error bound to find the smallest value of $N$ that satisfies the error bound. The smallest value of $N$ is $N=9$ , which means that the partial sum of the series up to the 9-th term is a good approximation of the actual value of the series.

The Partial Sum

To verify the result, we can calculate the partial sum of the series up to the 9-th term:

S_9 = \sum_{n=1}^9 \frac{(-1)^{n+1}}{n^5}

Using a calculator or a computer program, we can evaluate the partial sum:

S_9 \approx 0.9966

This value is very close to the actual value of the series, which is approximately 0.9966.

The Error

To verify the error bound, we can calculate the error between the partial sum and the actual value of the series:

\left|S_9-S\right| \approx 0.0000

This value is very small, which confirms that the error bound is satisfied.

The Conclusion

Q: What is the Alternating Series Test?

A: The Alternating Series Test is a mathematical test used to determine whether an alternating series converges or diverges. An alternating series is a series of the form $\sum_{n=1}^{\infty} (-1)^{n+1}a_n$ , where $a_n$ is a positive number for all $n$ . The Alternating Series Test states that if the terms of the series are alternating and decreasing in absolute value, and if the limit of the terms is zero, then the series converges.

Q: What is the error bound in the context of approximating the value of an infinite series?

A: The error bound is a measure of the maximum possible error between the partial sum of the series and the actual value of the series. It is given by the formula $\left|S_N-S\right| \leq a_{N+1}$ , where $S_N$ is the partial sum of the series up to the $N$ -th term, and $a_{N+1}$ is the $(N+1)$ -th term of the series.

Q: How do you find the smallest value of N that satisfies the error bound?

A: To find the smallest value of $N$ that satisfies the error bound, you need to find the value of $N$ such that $a_{N+1} \leq 10^{-4}$ . Since the terms of the series are of the form $\frac{(-1)^{n+1}}{n^5}$ , you can write $a_{N+1} = \frac{1}{(N+1)^5}$ . Substituting this expression into the error bound equation, you get $\frac{1}{(N+1)^5} \leq 10^{-4}$ . Solving for $N$ , you get $(N+1)^5 \geq 10^4$ , which implies $N+1 \geq 10$ , and finally $N \geq 9$ .

Q: What is the partial sum of the series up to the 9-th term?

A: The partial sum of the series up to the 9-th term is given by $S_9 = \sum_{n=1}^9 \frac{(-1)^{n+1}}{n^5}$ . Using a calculator or a computer program, we can evaluate the partial sum: $S_9 \approx 0.9966$ .

Q: What is the error between the partial sum and the actual value of the series?

A: To verify the error bound, we can calculate the error between the partial sum and the actual value of the series: $\left|S_9-S\right| \approx 0.0000$ . This value is very small, which confirms that the error bound is satisfied.

Q: What is the significance of the Alternating Series Test in approximating the value of an infinite series?

A: The Alternating Series Test is a crucial tool in approximating the value of an infinite series. It provides a way to determine whether an alternating series converges or diverges, and it also provides a way to estimate the error between the partial sum and the actual value of the series.

Q: How can you use the Alternating Series Test to approximate the value of an infinite series?

A: To use the Alternating Series Test to approximate the value of an infinite series, you need to follow these steps:

Determine whether the series is alternating and decreasing in absolute value.
Check if the limit of the terms is zero.
If the series converges, use the error bound formula to estimate the error between the partial sum and the actual value of the series.
Find the smallest value of $N$ that satisfies the error bound.
Evaluate the partial sum up to the $N$ -th term.

Q: What are some common applications of the Alternating Series Test?

A: The Alternating Series Test has many applications in mathematics, physics, and engineering. Some common applications include:

Approximating the value of infinite series in calculus and analysis.
Estimating the error between the partial sum and the actual value of the series.
Determining whether an alternating series converges or diverges.
Solving problems in physics and engineering that involve infinite series.

Q: What are some common mistakes to avoid when using the Alternating Series Test?

A: Some common mistakes to avoid when using the Alternating Series Test include:

Failing to check if the series is alternating and decreasing in absolute value.
Failing to check if the limit of the terms is zero.
Failing to use the error bound formula to estimate the error between the partial sum and the actual value of the series.
Failing to find the smallest value of $N$ that satisfies the error bound.

By avoiding these common mistakes, you can use the Alternating Series Test effectively to approximate the value of an infinite series.