Problem 50 - Thomas Calculus

Question

In Exercises \(49 - 52 ,\) estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { 10 ^ { n } } $$

Step-by-Step Solution

Answer

The error in the sum approximation is less than \( \frac{1}{100000} \).

Step 1: Identify the Series Type

The given series \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{10^n} \) is an alternating series. Alternating series have terms that alternate in sign.

Step 2: Use the Alternating Series Test

The Alternating Series Test states that if \(|a_{n+1}| < |a_n|\) and \(\lim_{n \to \infty} a_n = 0\), the series converges. For this series, \(a_n = \frac{1}{10^n}\) meets both conditions.

Step 3: Calculate the First Four Terms

Calculate the first four terms of the series: \(a_1 = \frac{1}{10}\), \(a_2 = \frac{1}{100}\), \(a_3 = \frac{1}{1000}\), and \(a_4 = \frac{1}{10000}\).

Step 4: Find the Sum of the First Four Terms

The sum of the first four terms is \(a_1 - a_2 + a_3 - a_4 = \frac{1}{10} - \frac{1}{100} + \frac{1}{1000} - \frac{1}{10000}\). Convert to a common denominator and calculate for more accuracy.

Step 5: Estimate the Error Using \(a_5\)

The error in alternating series is less than or equal to the absolute value of the first omitted term, which is \(a_5 = \frac{1}{100000}\).

Step 6: Conclude the Error Estimation

The magnitude of the error in approximating the sum with the first four terms is at most \( \frac{1}{100000} \).

Key Concepts

Error Estimation

When dealing with alternating series, estimating the error of an approximation is crucial for understanding the accuracy of your calculations. In our exercise, we are asked to estimate the error when approximating the sum of an entire infinite series by using just the first four terms. This process is known as error estimation.

To determine the potential error:

• Identify the term immediately following the terms included in your approximation.
• In this case, after considering the first four terms, the next term is the fifth term.
• The absolute value of this fifth term gives us the upper bound of the error.

This method provides a simple yet effective way to measure the error, ensuring that your approximation remains accurate within a reasonable range.

Convergence

Convergence is a vital concept when studying infinite series, like the one in our problem. An infinite series converges if the sum of its terms approaches a finite limit as more and more terms are added.

In our case, we use the Alternating Series Test to check for convergence:

• Each term must be smaller in absolute value than the one preceding it. This is clearly true as each term is a fraction with successively larger powers of ten in the denominator.
• The terms must ideally approach zero as they progress to infinity, ensuring the series doesn't diverge.

If these conditions are met, the series converges, meaning the omission of some terms will not significantly alter the sum's value.

Alternating Series Test

The Alternating Series Test is a tool used to determine whether an infinite series converges. This test is highly applicable to our problem since it deals with an alternating series, where each term alternates in sign.

For a series to pass this test:

• The absolute value of each term must be decreasing. Simplified, \(|a_{n+1}| < |a_n|\).
• The limit of the terms as they tend toward infinity should be zero, \(\lim_{n \to \infty} a_n = 0\).

In the given series, both conditions are satisfied:

• The sequence \(\frac{1}{10^n}\) ensures that each term is smaller than the last.
• As n increases, \(\lim_{n \to \infty} \frac{1}{10^n} = 0\).

Hence, the series is convergent by the Alternating Series Test.

Infinite Series

An infinite series is a sum of an infinite sequence of numbers. The sequence could potentially extend without end and is usually represented by a summation notation like \(\sum_{n=1}^{\infty} a_n\).

Understanding infinite series involves:

• Recognizing the pattern of the series, which could be arithmetic, geometric, or even alternating, as in our case.
• Using tests like the Alternating Series Test to determine convergence, assuring us that the series sums to a finite value.

When focusing on infinite alternating series, such as the one we have, remember:

• Terms alternate in sign, offering a unique way to check convergence.
• While practical computations only involve finite terms, the behavior of the series as it extends to infinity provides essential insights into its nature.

Thus, grasping infinite series concepts allows for comprehensive understanding and manipulation beyond finite arithmetic.