Question

Estimating the Remainder of an Alternating Series Consider the alternating series $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} $$ Use the remainder estimate to determine a bound on the error \(R_{10}\) if we approximate the sum of the series by the partial sum \(S_{10}\)

Short answer

The error \( R_{10} \) is bounded by \( \frac{1}{121} \).

Step-by-step solution

Understand the Problem

The series in question is \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} \), which is an alternating series. Our task is to estimate the error made by approximating the sum of this series using its first 10 terms, denoted by \( S_{10} \).

Recall the Alternating Series Test

The Alternating Series Test tells us that for an alternating series \( \sum_{n=1}^{\infty} (-1)^{n} a_n \), where \( a_n > 0 \), is convergent if the sequence \( a_n \) is decreasing and the limit of \( a_n \) as \( n \) approaches infinity is 0. Both conditions are satisfied for \( a_n = \frac{1}{n^2} \).

Use the Remainder Estimate for Alternating Series

The error \( R_n \) when estimating the sum of an alternating series by its first \( n \) terms is less than or equal to the absolute value of the first unused term. That is, \( |R_n| \leq a_{n+1} \). Thus, \( |R_{10}| \leq a_{11} = \frac{1}{11^2} \).

Calculate the Error Bound

Now we calculate \( a_{11} = \frac{1}{11^2} \). This equals \( \frac{1}{121} \). Thus, \( |R_{10}| \leq \frac{1}{121} \).

Conclude with the Error Bound

We conclude that the error in approximating the sum of the series by the first 10 terms is less than or equal to \( \frac{1}{121} \). This provides an effective upper bound for \( R_{10} \).

Key concepts

Remainder Estimate

When dealing with alternating series, the remainder estimate provides a way to understand how accurate a partial sum approximation is. For the series \( \sum_{n=1}^{\infty} (-1)^{n+1}a_n \), the remainder \( R_n \), which is the error or difference between the actual infinite sum and the partial sum \( S_n \), can be bounded. This is done by the absolute value of the first neglected term.

In simpler terms, when you stop summing at a certain point, the remainder estimate assures that the error will not be larger than the next term in the sequence you haven't included. For example, if you stop at the 10th term, the error can be no greater than the 11th term. This is particularly useful because it allows us to have a calculated comfort around how accurate our approximation is.

Convergence

Convergence is an essential concept in understanding infinite series. An infinite series is said to converge if the sum approaches a specific value as more and more terms are added. For alternating series like \( \sum_{n=1}^{\infty} (-1)^{n+1}a_n \), we often use the Alternating Series Test to determine convergence. This test requires two conditions:

• The terms \( a_n \) must be decreasing.
• \( a_n \) must approach 0 as \( n \) goes to infinity.

If both conditions are satisfied, like in the given series where \( a_n = \frac{1}{n^2} \), then the series converges. This means, as you add more and more terms, your partial sums get closer to a specific sum, ensuring that our calculations using partial sums can be trusted within an error bound.

Error Bound

The error bound in an alternating series gives a measure of how much the actual sum might differ from the sum of its first few terms. It is a handy tool because it provides direct information on the accuracy of a truncated series. In the case of the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} \), we use the formula for the remainder estimate: \(|R_n| \leq a_{n+1}\).

Specifically, for the first 10 terms, the error bound is \( |R_{10}| \leq \frac{1}{11^2} \).
Now, calculating gives us \( |R_{10}| \leq \frac{1}{121} \). This result tells us the maximum error we can expect between the estimated sum using the first 10 terms and the true sum of the series.

Partial Sum

A partial sum is the sum of the first few terms in an infinite series. Applying this concept to our series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} \), the partial sum \( S_{10} \) specifically denotes the sum of the first 10 terms.

Calculating a partial sum gives a finite approximation to an otherwise infinite process.
This is useful because it simplifies complex mathematical scenarios into manageable calculations. However, it is crucial to remember that a partial sum will only be an approximation; hence, knowing the error bound or remainder estimate becomes vital to judge how close \( S_{10} \) is to the actual infinite sum. Partial sums like \( S_{10} \) serve as building blocks in understanding the behavior of infinite series.