Problem 36 - Calculus: Early Transcendentals

Question

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{3^n + 4^n} \)

Step-by-Step Solution

Answer

The sum is approximately 0.186 with an error of 0.002.

Step 1: Identify the Series Type

The given series is \( \sum_{n=1}^{\infty} \frac{1}{3^n + 4^n} \). This is an infinite series where each term is given by \( a_n = \frac{1}{3^n + 4^n} \). We are asked to approximate the sum of the series using the sum of the first 10 terms, indicating we need to find a partial sum.

Step 2: Compute the First 10 Terms

Calculate the first 10 terms of the series: \( a_1 = \frac{1}{3^1 + 4^1} = \frac{1}{7} \), \( a_2 = \frac{1}{3^2 + 4^2} = \frac{1}{25} \), and so forth up to \( a_{10} = \frac{1}{3^{10} + 4^{10}} \).

Step 3: Approximate the Sum of the First 10 Terms

Sum the first 10 terms: \( S_{10} = a_1 + a_2 + ... + a_{10} \). Using a calculator, find each term's value and sum them, yielding \( S_{10} \approx 0.186 \).

Step 4: Estimate the Error Using the Tail Estimation

To estimate the error, use the n-th term test for convergence and note the series terms decrease in size similarly to a geometric series with ratio \( \left( \frac{3}{4} \right)^n \). Use the formula for the tail approximation of an infinite series: the error \( R_N \approx a_{N+1} \). Thus, \( \text{Error} \approx \frac{1}{3^{11} + 4^{11}} \approx 0.002 \).

Step 5: Conclusion

The approximate sum of the series using the first 10 terms is \( 0.186 \) with an estimated error of \( 0.002 \).

Key Concepts

Partial Sum

When working with infinite series, calculating a partial sum is a practical way to approximate the series' total sum. In this context, a partial sum is the sum of a finite number of terms from the series. For instance, in the exercise given,

1. we take the first 10 terms of the series given by \( \sum_{n=1}^{\infty} \frac{1}{3^n + 4^n} \).
2. Once we identify and sum these first 10 terms, it allows us to probe the behavior of the series and gain an approximation of the sum of the infinite series.
3. This is a crucial step because calculating the full infinite series directly is not feasible.

By using only these initial terms, one can get a approximate picture of the series' sum, denoted as \( S_{10} \), which in our solution, was approximately \( 0.186 \).

This approach aids in estimating sums without needing to evaluate an endless number of terms.

Error Estimation

When approximating an infinite series with a partial sum, some difference, known as the error, remains between this approximation and the exact sum. Error estimation helps us understand the accuracy of the partial sum.

• In the series in question, we employ the error estimation by calculating the next term after the 10th, namely \( a_{11} = \frac{1}{3^{11} + 4^{11}} \), as an approximation of error.
• Since this series resembles a geometric series, the subsequent terms decrease proportionally.

Thus, the error is quite small, approximately \( 0.002 \) in this solution. By utilizing error estimation, we gain a sense of how close our partial sum is to the actual sum, allowing for calculated approximations.

Convergence Test

Before estimating the sum of an infinite series, verifying its behavior with a convergence test is essential to ensure it converges to a finite number.

This series looks like a geometric series, which provides a basis for convergence evaluation using the ratio test or comparison with known geometric series.

• If the terms decrease at a consistent rate—as seen here with terms like \( \left( \frac{3}{4} \right)^n \)—it's more likely that the series converges.
• For our specific series, observing that each subsequent term decreases in value similar to a geometric pattern helped confirm convergence.

Convergence tests are indispensable in mathematics for gauging whether approximations like partial sums truly reflect the infinite series' behavior.

Geometric Series

A geometric series is a type of series where each term after the first is found by multiplying the previous one by a constant called the common ratio. These series have easily identifiable patterns that simplify analysis.

• For example, in a simple geometric series \( a + ar + ar^2 + \ldots \), the sum can be calculated using the formula \( S = \frac{a}{1 - r} \).
• Even though our series is not purely geometric, it behaves similarly with a ratio \( \left( \frac{3}{4} \right)^n \).

This approximation of behavior facilitates using geometric series principles for error estimation and partial sum calculations. Understanding geometric series allows for a deeper comprehension of how to approach and simplify many infinite series problems.