Problem 4 - Active Calculus

Question

For the following alternating series, \(\sum_{n=1}^{\infty} a_{n}=1-\frac{1}{10}+\frac{1}{100}-\frac{1}{1000}+\ldots\) how many terms do you have to go for your approximation (your partial sum) to be within 1e-07 from the convergent value of that series?

Step-by-Step Solution

Answer

7 terms

Step 1 - Identify the general term of the series

The general term of the series can be written as \(a_n = \frac{(-1)^{n+1}}{10^{n-1}}\).

Step 2 - Apply the Alternating Series Error Estimation Theorem

The theorem states that for an alternating series, the error after \(n\) terms is less than or equal to the absolute value of the \((n+1)\)-th term, \(|a_{n+1}|\). Set this less than 1e-07: \(|a_{n+1}| < 1 \times 10^{-7}\).

Step 3 - Solve for n to find the smallest term that fits the error criterion

From the inequality \(\left| \frac{(-1)^{(n+1)}}{10^n} \right| < 1 \times 10^{-7}\), we get \(\frac{1}{10^n} < 1 \times 10^{-7}\). Solving for \(n\), we take logarithms to get: \(n > \log_{10}(10^7) = 7\).

Step 4 - Conclusion

Since \(n\) must be greater than 7, we need to include at least 7 terms. Therefore, the partial sum of the first 7 terms will be within the required error margin.

Key Concepts

Alternating Series

An alternating series is a series in which the signs of the terms alternate between positive and negative. It is typically written in the form \sum_{n=1}^{\infty} (-1)^{n} b_{n}, where \(b_{n}\) is a sequence of positive terms. One well-known example is the alternating harmonic series:\ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots.\ Alternating series are interesting because they often converge even when the similar series with all positive terms would diverge. This makes them useful in approximating functions and values. Moreover, the behavior of alternating series is governed by the Alternating Series Test, ensuring the series converges if the terms \(b_{n}\) are decreasing and approach zero.

Error Approximation

When working with alternating series, we often want to know how close a partial sum is to the full sum of the series. This is called error approximation. The Alternating Series Error Estimation Theorem helps us find this error. For an alternating series, the error after summing the first \(n\) terms is less than or equal to the absolute value of the next term, or \(|a_{n+1}|\). This means that if you stop at the nth term, your error is no more than the size of the (n+1)th term. Understanding error approximation is crucial for practical applications, because it tells us how good our approximation is.

Convergence of Series

The convergence of a series refers to whether the sum of its terms approaches a finite limit as more terms are added. For alternating series, the Alternating Series Test (or Leibniz's criterion) is a simple way to check this. An alternating series \sum_{n=1}^{\infty} (-1)^{n} b_{n} converges if two conditions are met: (1) The magnitude of the terms \(b_{n}\) decreases monotonically (each term is smaller than the one before). (2) The limit of \(b_{n}\) as \(n\) approaches infinity is zero. When these conditions are satisfied, the series is guaranteed to converge, and you can use partial sums to approximate its value accurately.

Logarithmic Inequality

Logarithmic inequality involves using the properties of logarithms to solve inequalities. In the context of this exercise, we need to solve \(\frac{1}{10^n} < 1 \times 10^{-7}\). To solve for \(n\), we can use logarithms. First, rewrite the inequality as \(10^{-n} < 10^{-7}\). Taking the logarithm base 10 of both sides, we obtain: \(-n < -7\) or simply \(n > 7\). This tells us that \(n\) must be greater than 7 for the partial sum to be within the error margin. Thus, logarithmic inequality helps in finding out how many terms are needed to meet a specific accuracy when working with series.