Taylor Series Error Bound Find The Number Of Terms N

The Taylor series is a powerful tool for approximating the value of a function at a specific point using its derivatives at another point. However, the approximation is not perfect, and there is always an associated error. Understanding the Taylor series error bound allows us to determine the accuracy of our approximation and, crucially, find the minimum number of terms, n, required to achieve a desired level of accuracy. This article will delve into the Taylor series error bound, exploring its mathematical foundation, practical application, and how to calculate the necessary number of terms for a given accuracy.

Understanding the Taylor Series and Remainder

Before diving into the error bound, let's briefly recap the Taylor series itself. The Taylor series of a function f(x) about the point a is given by:

f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... + f^(n)(a)(x-a)^n/n! + ...

Where:

• f'(a), f''(a), f'''(a), ... f^(n)(a) are the first, second, third, and nth derivatives of f(x) evaluated at x = a.
• n! denotes the factorial of n.
• a is the point about which the series is expanded (the center of the Taylor series).

The Taylor polynomial of degree n, T_n(x), is the partial sum of the Taylor series up to the nth term:

T_n(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + ... + f^(n)(a)(x-a)^n/n!

This Taylor polynomial provides an approximation of f(x). The difference between the actual function value, f(x), and the Taylor polynomial, T_n(x), is the remainder or error, denoted by R_n(x):

R_n(x) = f(x) - T_n(x)

The Taylor series error bound focuses on providing a bound for the absolute value of this remainder.

The Taylor's Remainder Theorem (Error Bound)

Taylor's Remainder Theorem provides a way to estimate the maximum possible error in approximating f(x) with its Taylor polynomial T_n(x). There are several forms of the remainder, but the most common and useful for our purposes is the Lagrange form of the remainder:

|R_n(x)| <= M |(x-a)^(n+1)| / (n+1)!

Where:

• |R_n(x)| is the absolute value of the remainder, representing the error.
• M is an upper bound for the absolute value of the (n+1)th derivative of f(x) on the interval between a and x. In other words, we need to find a value M such that |f^(n+1)(t)| <= M for all t between a and x. This is the crucial step and often the most challenging.
• x is the point at which we are approximating the function value.
• a is the center of the Taylor series.
• n is the degree of the Taylor polynomial (the number of terms used, minus one).
• (n+1)! is the factorial of (n+1).

Key Idea: The Taylor's Remainder Theorem tells us that the error is bounded by a term involving the (n+1)th derivative, the distance between x and a raised to the power of (n+1), and the factorial of (n+1). The M term is the upper bound of the (n+1)th derivative and significantly influences the error bound.

Finding M: The Upper Bound of the (n+1)th Derivative

Determining the value of M is often the most challenging part of using the Taylor series error bound. It requires finding an upper bound for the absolute value of the (n+1)th derivative, |f^(n+1)(t)|, on the interval between a and x. Here's a breakdown of strategies:

1. Direct Calculation and Analysis:

   • Calculate the (n+1)th derivative of f(x), f^(n+1)(x).
   • Analyze the behavior of |f^(n+1)(x)| on the interval [a, x] (or [x, a] if x < a).
   • Find the maximum value of |f^(n+1)(x)| on this interval. This value serves as M.

   Example: Let's say f(x) = sin(x), a = 0, and we're approximating sin(0.5) using a Taylor polynomial. Suppose we want to find M for n = 2 (a second-degree Taylor polynomial).

   • f'(x) = cos(x)
   • f''(x) = -sin(x)
   • f'''(x) = -cos(x) (This is our (n+1)th derivative)

   We need to find the maximum of |-cos(x)| on the interval [0, 0.5]. Since cosine is a decreasing function on this interval and |cos(0)| = 1, and |-cos(x)| is always less than or equal to 1, we can take M = 1.

2. Using Known Inequalities:

   • Leverage known inequalities to bound the derivative. For example:

     • |sin(x)| <= 1 for all x.
     • |cos(x)| <= 1 for all x.
     • e^x <= e^b for x <= b

   Example: Consider f(x) = e^(-x), a = 0, and we are approximating e^(-0.2). We are interested in finding an M for n=3.

   • f'(x) = -e^(-x)
   • f''(x) = e^(-x)
   • f'''(x) = -e^(-x)
   • f''''(x) = e^(-x) (This is our (n+1)th derivative).

   We need to find the maximum of |e^(-x)| on the interval [0, 0.2]. Since e^(-x) is a decreasing function on this interval, its maximum occurs at x = 0. Therefore, e^(-0) = 1. So, M = 1.

3. Considering Monotonicity and Concavity:

   • Determine if the absolute value of the derivative is increasing, decreasing, or has a maximum within the interval [a, x].
   • If it's monotonic (either increasing or decreasing), the maximum value will occur at one of the endpoints (a or x).
   • If it has a maximum within the interval, you might need to use calculus techniques (finding critical points) to locate the maximum.

4. Bounding with a Simpler Function:

   • Sometimes, it's helpful to find a simpler function that bounds the absolute value of the derivative. For example, if |f^(n+1)(x)| <= g(x) on the interval [a, x], and you can easily find the maximum of g(x) on that interval, you can use that maximum as M.

Important Considerations for finding M:

• Interval: The interval [a, x] is crucial. The maximum value of the derivative must be found within this interval. A value outside this interval is not a valid M.
• Upper Bound: M is an upper bound, meaning it's a value greater than or equal to the absolute value of the derivative on the interval. It doesn't have to be the exact maximum, just a value that is definitely no smaller than the maximum. Overestimating M is acceptable; underestimating it is not, as it will lead to an inaccurate error bound.
• Practicality: Sometimes, finding the absolute smallest possible M is very difficult. In practice, it's often sufficient to find a reasonably good upper bound that makes the calculation manageable.

Finding the Number of Terms, n, for a Desired Accuracy

The primary application of the Taylor series error bound is to determine the number of terms, n, needed in the Taylor polynomial to achieve a desired level of accuracy. Here's the process:

1. Determine the Desired Accuracy (Error Tolerance):

   • Decide how accurate you need the approximation to be. This is typically expressed as a maximum allowable error, often denoted by ε (epsilon). You want |R_n(x)| <= ε.

2. Find an Expression for the Error Bound:

   • Use the Taylor's Remainder Theorem: |R_n(x)| <= M |(x-a)^(n+1)| / (n+1)!
   • Determine the appropriate M value as described above.

3. Set Up the Inequality:

   • Set the error bound less than or equal to the desired accuracy:

     M |(x-a)^(n+1)| / (n+1)! <= ε

4. Solve for n:

   • This is the most challenging step, as it involves solving an inequality with a factorial. There's no general algebraic solution for n in this type of inequality. The common approaches are:

     • Trial and Error (Iteration): Start with a small value of n (e.g., n = 0, 1, 2, ...). Calculate the error bound for each value of n until the error bound is less than or equal to ε. This is often the most practical approach. You can use a calculator or computer to automate this process.

     • Approximations and Estimation: In some cases, you might be able to use approximations to simplify the factorial or other terms in the inequality to get a rough estimate of n. However, always verify the result with trial and error to ensure it meets the desired accuracy. Stirling's approximation can be helpful for large n:

       n! ≈ √(2πn) (n/e)^n

       However, using Stirling's approximation within an inequality still often requires iterative solving.

5. Choose the Smallest Integer n:

   • Once you find a value of n that satisfies the inequality, choose the smallest integer greater than or equal to that value. This ensures that the error is within the desired tolerance.

Example: Approximating cos(0.1) using the Maclaurin Series (Taylor Series centered at a=0)

Let's approximate cos(0.1) using its Maclaurin series and determine the number of terms, n, required to achieve an accuracy of ε = 0.0001 (0.01%).

1. Maclaurin Series for cos(x):

   cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

2. Desired Accuracy: ε = 0.0001

3. Error Bound:

   |R_n(x)| <= M |x^(n+1)| / (n+1)!

   Here, x = 0.1 and a = 0. We need to find M.

4. Finding M:

   The derivatives of cos(x) are cos(x), -sin(x), -cos(x), sin(x), and they repeat in a cycle. Therefore, |f^(n+1)(x)| will always be either |cos(x)| or |sin(x)|. Since |cos(x)| <= 1 and |sin(x)| <= 1 for all x, we can take M = 1.

5. Setting up the Inequality:

   1 * |(0.1)^(n+1)| / (n+1)! <= 0.0001

   |(0.1)^(n+1)| / (n+1)! <= 0.0001

6. Solving for n (Trial and Error):

   • n = 0: (0.1)^1 / 1! = 0.1 > 0.0001
   • n = 1: (0.1)^2 / 2! = 0.005 > 0.0001
   • n = 2: (0.1)^3 / 3! = 0.00001666... < 0.0001

   Therefore, n = 2 is sufficient. Notice that the Maclaurin series for cosine only contains even powers of x. When n = 2, the Taylor polynomial includes terms up to x^2. This implies that the first three terms of the series (1 - x^2/2!) provide the desired accuracy.

Conclusion for the Example: We need n = 2, meaning we need the terms up to x^2 in the Maclaurin series, to approximate cos(0.1) with an accuracy of 0.0001. The approximation is:

cos(0.1) ≈ 1 - (0.1)^2 / 2! = 1 - 0.005 = 0.995

The actual value of cos(0.1) is approximately 0.995004165. The error is about 0.000004165, which is less than our desired accuracy of 0.0001.

Practical Considerations and Common Mistakes

• Choosing the Correct Center (a): The choice of the center a significantly impacts the convergence and accuracy of the Taylor series. Choose a close to the value x at which you're approximating the function. This generally leads to faster convergence and smaller errors.
• Finding M Accurately: Underestimating M leads to an incorrect error bound, providing a false sense of accuracy. Always ensure M is a valid upper bound for the (n+1)th derivative on the interval.
• Interval of Convergence: The Taylor series may only converge for certain values of x. Make sure x is within the interval of convergence of the Taylor series. If it's not, the error bound may not be valid.
• Computational Errors: Be careful with calculations, especially when dealing with factorials and exponents. Use a calculator or computer to avoid errors.
• Understanding the "n" Value: Remember that n represents the degree of the Taylor polynomial. The Taylor polynomial includes terms from degree 0 up to degree n, so it has (n+1) terms.
• Alternating Series Test: If the Taylor series is an alternating series, the Alternating Series Test provides a simpler error bound: the error is less than or equal to the absolute value of the first neglected term. However, this test only applies to alternating series that satisfy specific conditions.

Conclusion

The Taylor series error bound is a critical tool for understanding and controlling the accuracy of Taylor series approximations. By carefully determining the upper bound M of the (n+1)th derivative and applying the Taylor's Remainder Theorem, we can calculate the maximum possible error. This allows us to find the number of terms, n, required to achieve a desired level of accuracy. While finding M and solving for n can sometimes be challenging, the process provides valuable insight into the behavior of the Taylor series and its limitations. Mastering the Taylor series error bound is essential for anyone using Taylor series for numerical approximation and analysis. Using the trial-and-error method is often the most practical approach for finding the suitable number of terms, n, especially when dealing with factorials. Remember to verify the solution to ensure it is within the desired error tolerance.