Question

The following alternating series converge to given multiples of \(\pi .\) Find the value of \(N\) predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(N\) for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, \(\pi=3.141592653589793 .\)[T] \(\frac{\pi}{4}=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 n+1}\), error \(<0.0001\)

Short answer

The minimum \( N \) is 5000; the approximation of \( \pi \) is found by calculating the sum up to the 5000th term and multiplying by 4.

Step-by-step solution

Understand the given series

The alternating series given is \( \frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} \). This series is known as the Leibniz formula for \( \pi \). We want to find the smallest \( N \) such that the Nth partial sum approximates \( \frac{\pi}{4} \) to within an error of 0.0001.

Applying the Alternating Series Remainder Estimate

For an alternating series, the remainder (which is the error of approximation by the Nth partial sum) is less than the absolute value of the first omitted term. For this series, the remainder after the \( N \)th term is \( \left| R_N \right| = \left| \frac{(-1)^{N+1}}{2N+1} \right| = \frac{1}{2N+1} \). We want this to be less than 0.0001.

Solve the inequality for N

We set up the inequality \( \frac{1}{2N+1} < 0.0001 \) and solve for \( N \). Multiplying both sides by \( 2N+1 \) gives \( 1 < 0.0001(2N+1) \). Simplifying, we get \( 2N+1 > 10000 \). Further simplifying, we find \( 2N > 9999 \), so \( N > 4999.5 \). Since \( N \) must be an integer, we take \( N = 5000 \).

Calculate the partial sum for N = 5000

We calculate the partial sum \( S_{5000} = \sum_{n=0}^{5000} \frac{(-1)^{n}}{2n+1} \). The computation will give the approximate value for \( \frac{\pi}{4} \). This value can then be multiplied by 4 to approximate \( \pi \).

Verify the error estimate

Recalculate the remainder with \( N=5000 \). The next term after \( N = 5000 \) is \( \frac{1}{10001} \approx 0.00009999 \), which is indeed less than 0.0001, confirming \( N = 5000 \) meets the criteria.

Key concepts

Leibniz Formula for Pi

The Leibniz formula for pi is a fascinating mathematical series used to approximate the value of pi. It expresses pi divided by 4 as an infinite series:

• \( \frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} \)

This formula represents an alternating series where the terms alternately add and subtract. Each term consists of a fraction with a numerator of \((-1)^{n}\) and a denominator that increases progressively by 2 with each term. The series converges, meaning it approaches a specific value, allowing us to approximate \(\pi\) by calculating a finite number of terms. This makes it a useful tool for getting close to \(\pi\) without going through endless calculations.

Alternating Series Remainder Estimate

When working with alternating series, an important concept is the remainder estimate, which gives us a handle on the error involved when approximating a sum using a finite number of terms. The remainder in an alternating series, such as the Leibniz formula, can be calculated by taking the absolute value of the next term not included in the partial sum.

• For the series \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} \), the remainder after \(N\) terms, \( |R_N| \), is given by \( \frac{1}{2N+1} \).

To ensure our approximation reaches a desired accuracy, we set this remainder less than a given error tolerance, for instance, \(0.0001\). Solving for \(N\) gives us the smallest number of terms needed to achieve the precision we want.
This approach not only helps in determining how many terms are necessary but also assures the result meets specific standards of accuracy.

Partial Sum Approximation

The partial sum in the context of series is simply the sum of the first \(N\) terms. It gives us an approximation of the series’ total sum. For instance, with the Leibniz formula, calculating the partial sum \(S_N\) involves summing the first \(N\) terms to approximate \(\frac{\pi}{4}\):

• \( S_N = \sum_{n=0}^{N} \frac{(-1)^{n}}{2n+1} \)

Once you have \(S_N\), you can multiply by 4 to get an approximation of \(\pi\).
This method is useful because it offers flexibility; you can choose \(N\) based on how precise you want your approximation to be. The idea here is selecting \(N\) large enough so that the alternating series remainder estimate shows the error is below your acceptable level.

Series Convergence

Series convergence refers to where the partial sums of a series increasingly approach a limit. For an alternating series, like the Leibniz series for \(\pi\), determining convergence is crucial to ensuring the series can be trusted in approximations. An alternating series will converge if the absolute values of the terms decrease steadily toward zero.

• The terms in the Leibniz series \( \frac{1}{2n+1} \) affirm this by diminishing as \(n\) grows larger.

An important property of a converging series is that it eventually settles to a number, in this case, \(\frac{\pi}{4}\), implying that adding enough terms will yield a value progressively akin to the target value. This is why convergence is vital in validating both the accuracy and reliability of these approximations.