Problem 54 - Thomas Calculus

Question

In Exercises \(53 - 56 ,\) determine how many terms should be used to estimate the sum of the entire series with an error of less than \(0.001 .\) $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { n } { n ^ { 2 } + 1 } $$

Step-by-Step Solution

Answer

31 terms are needed for an error less than 0.001.

Step 1: Identify the Series and Error Bound

The series given is \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n^2 + 1}\). You need to find how many terms are needed to have a sum with an error of less than 0.001. This is an alternating series.

Step 2: Use Alternating Series Remainder Theorem

The Alternating Series Remainder Theorem states that the absolute error \(R_n\) in the sum of the first \(n\) terms of an alternating series is less than or equal to the absolute value of the first neglected term. That is, \(R_n = |S - S_n| \leq |a_{n+1}|\). You want this \(|a_{n+1}| < 0.001\).

Step 3: Define the General Term

For the series \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n^2 + 1}\), the general term is \(a_n = \frac{n}{n^2 + 1}\). To find the smallest \(n\) such that \(|a_{n+1}| < 0.001\), examine \(|a_{n+1}| = \frac{n+1}{(n+1)^2 + 1}\).

Step 4: Solve the Inequality

Solve \(\frac{n+1}{(n+1)^2 + 1} < 0.001\). This implies solving \(n+1 < 0.001((n+1)^2 + 1)\). After simplification, check values of \(n\) to find the smallest integer for which this inequality holds.

Step 5: Numerical Verification

Through trial and error or computation, determine that this inequality holds for \(n \approx 31\). Hence, you would need 31 or more terms for the error to be less than 0.001.

Key Concepts

Remainder Theorem in Alternating Series

The Remainder Theorem plays a pivotal role in understanding the behavior of alternating series. When dealing with such a series, the Alternating Series Remainder Theorem gives us a convenient way to estimate the magnitude of the error between the actual infinite sum of the series and the partial sum of the first few terms.
The theorem states that in an alternating series of the form:

• \(S = \sum_{n=1}^{\infty} (-1)^{n+1} a_n\)

the absolute error \(R_n\) in the sum of the first \(n\) terms is less than or equal to the absolute value of the first term that you do not sum. Mathematically, this is written as:
\[ R_n = |S - S_n| \leq |a_{n+1}| \]
where \(S_n\) is the sum of the first \(n\) terms. This can be exceptionally useful because it tells you the maximum possible error after summing a certain number of terms. By ensuring \(|a_{n+1}|\) is smaller than a desired error threshold, like 0.001, you can determine a sufficient number of terms to achieve this accuracy.

Convergence in Alternating Series

In mathematics, understanding whether or not a series converges is crucial. An alternating series potentially converges if its terms decrease in absolute value and approach zero as \(n\) becomes very large. For an alternating series like:

• \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n^2 + 1}\)

Ensuring convergence is slightly different than other series. The conditions for the Alternating Series Test (sometimes called the Leibniz Test) are:

• The terms \(a_n = \frac{n}{n^2 + 1}\) must be positive, which they are since the fraction is always positive.
• The sequence \(a_n\) must be decreasing. Although it might not be immediately obvious, \(\frac{n}{n^2 + 1}\) decreases as \(n\) increases, since the denominator grows faster than the numerator.
• The limit of \(a_n\) as \(n\) approaches infinity must be 0. Indeed, \(\lim_{n \to \infty} \frac{n}{n^2 + 1} = 0\).

If these conditions hold true, the series converges. Thus, with these checks, the series from the exercise is confirmed to converge.

Error Estimation in Series

Error estimation in series is a fundamental concept when we want a partial sum to approximate an infinite series. When dealing with alternating series, accurately gauging the error or the difference between our finite approximation and the true infinite sum is simplified via the Alternating Series Remainder Theorem.
When we compute terms in the series \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n^2 + 1}\), determining how big that error remains can be crucial for practical purposes. With the rule \(R_n \leq |a_{n+1}|\), finding \(n\) such that \(|a_{n+1}| < 0.001\) ensures the overall error in our estimation of the series sum is within specified bounds.
In this exercise, solving the inequality \(\frac{n+1}{(n+1)^2 + 1} < 0.001\) is the key. With some calculations, it's determined that when \(n = 31\), \(\frac{32}{32^2 + 1} < 0.001\), confirming that 31 terms are needed to make sure the error is less than 0.001. This concrete limit assists in practical applications where approximations are necessary, ensuring accurate estimations with confidence.