Problem 52 - Thomas Calculus

Question

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. $$ \frac{1}{1+t}=\sum_{n=0}^{\infty}(-1)^{n} t^{n}, \quad 0 < t < 1 $$

Step-by-Step Solution

Answer

The error magnitude is \( t^4 \).

Step 1: Identify the Series

The expression given is a geometric series: \( \sum_{n=0}^{\infty} (-1)^{n} t^{n} = \frac{1}{1+t} \). This type of series is characterized by a common ratio. The terms given are appropriate for small values of \( t \).

Step 2: Compute the First Four Terms

Calculate the first four terms of the series: \(\begin{align*}&\text{First term (n=0): } (-1)^0 t^0 = 1 ewline&\text{Second term (n=1): } (-1)^1 t^1 = -t ewline&\text{Third term (n=2): } (-1)^2 t^2 = t^2 ewline&\text{Fourth term (n=3): } (-1)^3 t^3 = -t^3 ewline\end{align*}\)

Step 3: Approximate Partial Sum

The sum of the first four terms is: \( 1 - t + t^2 - t^3 \). This is the approximation of the series using the first four terms alone.

Step 4: Calculate the Error Term

In a geometric series, the error when truncating after the nth term is the magnitude of the next term. Here, the next term (n=4) is \((-1)^4 t^4\), or simply \(t^4\). Thus, the error is \(|t^4|\).

Step 5: Express the Magnitude of the Error

The magnitude of the error is determined by the size of only the fifth term, which is \(t^4\). Hence, the magnitude of the error in using the first four terms to approximate the infinite series is \(t^4\).

Key Concepts

Series Approximation

In mathematics, approximating a series means estimating the sum of an infinite number of terms using only a finite number of those terms. This is particularly useful when we deal with geometric series, like the one in the given exercise: \[ \sum_{n=0}^{\infty} (-1)^{n} t^{n} = \frac{1}{1+t} \] This formula provides an exact representation for the infinite series, but in practical applications, it's often sufficient to use only the first few terms to get a very close approximation of the sum. In the exercise, the first four terms are used, specifically:

• 1 (for n=0)
• -t (for n=1)
• t^2 (for n=2)
• -t^3 (for n=3)

When these terms are added together, we get the partial sum used for approximation: \[ 1 - t + t^2 - t^3 \] This makes our calculations more manageable while still retaining much of the series' accuracy.

Error Estimation

Error estimation is the process of quantifying how accurate or close our approximation is to the actual infinite series. In geometric series, the error when using a finite number of terms is often expressed by the magnitude of the next unused term.In the exercise concerning the function \( \frac{1}{1+t} \), after computing the sum of the first four terms, the next term is \((-1)^4 t^4\), simplifying to \(t^4\). This term represents the error because it is the first term not included in our approximation. The formula looks like this:\[ \text{Error magnitude} = | t^4 | \]Taking the absolute value ensures that we only consider the size of the error, not its sign. In practical terms, if \( t \) is small, then \( t^4 \) will also be small, making our approximation highly accurate.

Partial Sum

A partial sum of an infinite series, like the geometric series we're dealing with, is when we add together only a certain number of initial terms. This gives us a simplified version of the full series which can be used to estimate the entire sum. For the series \( \sum_{n=0}^{\infty} (-1)^{n} t^{n} \), the partial sum using the first four terms is: \[ 1 - t + t^2 - t^3 \] This expression acts as an approximation to the infinite sum \( \frac{1}{1+t} \), offering a balance between computational simplicity and precision. It captures the essence of the series over the range of \( t \) values of interest, making complex calculations quicker and easier to manage. As a result, working with partial sums is a basic yet powerful technique in dealing with series in real-world applications.