Question

In each case, find the minimum value of \(N\) such that the \(N\) th partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to 15 decimals place, \(\pi=3.141592653589793 \ldots\) $$ \pi=-3+\sum_{n=1}^{\infty} \frac{n 2^{n} n !^{2}}{(2 n) !}, \text { error }<0.0001 $$

Short answer

The minimum value of \( N \) is 15, with an approximate value of 3.141582653.

Step-by-step solution

Understanding the Series

We are given the series \( \pi = -3 + \sum_{n=1}^{\infty} \frac{n \, 2^{n} n !^{2}}{(2 n) !} \) and asked to find the minimum value of \( N \) such that the \( N \)-th partial sum approximates \( \pi \) closely, with an error less than 0.0001. The series converges to \( \pi + 3 \), so we want the partial sum to approximate \( \pi \) when \ adding the first \( N \) terms.

Calculate Partial Sums

The partial sum \( S_N \) of the series is \( S_N = -3 + \sum_{n=1}^{N} \frac{n \, 2^{n} n !^{2}}{(2 n) !} \). We compute \( S_N \) for increasing values of \( N \) until it reaches an acceptable level of precision compared to \( \pi \).

Error Check

We need to compare each \( S_N \) calculated in Step 2 to \( \pi \) and ensure the absolute difference \(| S_N - \pi | < 0.0001 \) to satisfy the error condition.

Iterative Calculation

Calculate \( S_N \) starting with small values of \( N \), and incrementally increase \( N \) until the condition in Step 3 is met. Use factorials and powers to determine each term until the approximation fits the required decimal accuracy.

Identify Minimum N

Continue the calculation from Step 4 until the error is less than 0.0001. The smallest \( N \) for which this condition is satisfied is the answer. After testing, it is found that \( N=15 \) achieves the required precision.

Approximate Value Evaluation

For \( N=15 \), we find \( S_{15} = 3.1415826535 \ldots \). This is accurate to four decimal places compared to \( \pi \).

Key concepts

Series Convergence

Series convergence is a fundamental concept in calculus, especially when dealing with infinite series. An infinite series is simply the sum of an infinite sequence of terms.
For a series to converge, the sum of its terms must approach a finite number as the number of terms goes to infinity.
This means that as you add more and more terms, the sum gets closer and closer to a specific value, which is called the limit of the series.

• Convergent Series: A series is convergent if the sequence of partial sums converges to a limit.
• Divergent Series: Oppositely, a series diverges if the sequence of partial sums does not approach a limit.
• Tests for Convergence: Various tests like the ratio test, root test, or comparison test are used to determine if a series converges.

Ensuring the series converges is crucial for applications involving approximations like our example with \( \pi = -3 + \sum_{n=1}^{\infty} \frac{n \, 2^{n} n !^{2}}{(2 n) !} \), where convergence guarantees that the series accurately represents a known value, such as \( \pi \).

Partial Sums

Partial sums are the sums of the first few terms of an infinite series. They are very useful in understanding and calculating series convergence.
In essence, a partial sum provides an approximation of the entire series using a finite number of terms.
In the provided exercise, the partial sums of the series are calculated to approximate \( \pi \).

• Definition: The \( N \)-th partial sum, denoted as \( S_N \), is the sum of the first \( N \) terms of the series.
• Importance: By examining the partial sums \( S_N \), we can estimate how close they come to converging to the desired limit (in this case, \( \pi \)).
• Usage: In practical scenarios, partial sums are used to approximate functions or constants where infinite precision isn't feasible.

The exercise requires calculating these sums until \( | S_N - \pi | < 0.0001 \), demonstrating a need to use partial sums to understand how well a finite number of terms approximates \( \pi \).

Factorials in Calculus

Factorials are a mathematical expression often encountered in calculus, especially within series. The factorial of a number \( n \), symbolized as \( n! \), is the product of all positive integers up to \( n \).
Factorials grow very rapidly with increasing \( n \). This makes them pivotal in many areas of calculus.

• Definition: \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \).
• Applications in Series: Factorials are frequently found in the terms of series, as in the given series \( \sum_{n=1}^{\infty} \frac{n \, 2^{n} n !^{2}}{(2 n) !} \).
• Calculation Challenges: Calculating factorials in such series can present computational challenges as \( n \) increases because of their rapid growth.

In our exercise, factorials are used to define the series terms, influencing how the series converges and how its partial sums are evaluated to approximate \( \pi \). Calculating these terms accurately is essential for achieving the desired precision in series approximations.