AP Calculus: Alternating Series Error Bound Theorem

AP Calculus: Alternating Series Error Bound Theorem

Alternating Series Error Bound Theorem

Provides a way to estimate the error when approximating the sum of an alternating series using its partial sums.

Alternating Series

A series of the form ∑ (-1)^n a_n = a_0 - a_1 + a_2 - a_3 + ... where a_n ≥ 0 for all n.

Convergence of an Alternating Series

The series converges if the terms are positive, decreasing, and approach zero.

Partial Sum (S_n)

The sum of the first n terms of the series.

Error (E_n)

The difference between the actual sum (S) and the partial sum (S_n).

Error Bound

The error in approximating the sum by the n-th partial sum is at most the absolute value of the (n+1)-th term of the series.

Mathematical Formulation of Error

|E_n| = |S - S_n| ≤ a_{n+1}.

Implication of the Theorem

It provides a straightforward way to estimate how close S_n is to the actual sum S.

Practical Use of Theorem

Useful in numerical methods and applications where an approximation is needed.

Convergence Assurance

Error decreases as more terms are included in the sum.

Example of Alternating Series

∑ (n=0 to ∞) (-1)^n / (n+1) = 1 - 1/2 + 1/3 - 1/4 + ...

Terms of Example Series

In the example series, a_n = 1/(n+1).

Checking Conditions of Series

Terms a_n must be positive, decreasing, and approach zero.

Calculating Partial Sum S_4

S_4 = 1 - 0.5 + 0.333 - 0.25 + 0.2 = 0.783.

Estimating the Error in Example

The next term a_5 = 1/6, which gives an upper bound for the error |E_4| ≤ 1/6.

Interval for Actual Sum

S ∈ (0.616, 0.949) based on the example of S_4.

Taylor Series Expansion

A way to approximate functions using infinite series.

Approximation of ln(1+x)

ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ... around x=0.

Calculating Partial Sum S_4 for ln(1.5)

S_4 = 0.5 - 0.125 + 0.04167 - 0.015625 ≈ 0.4000.

Error Estimation for ln(1.5)

The next term a_5 = 0.00625 allows for error estimation |E_4| ≤ 0.00625.

Estimating sin(x) with Taylor Series

sin(x) = x - x^3/3! + x^5/5! - ... around x=0.

Calculating Partial Sum S_3 for sin(0.5)

S_3 = 0.5 - 0.125/6 + 0.03125/120 ≈ 0.4794.

Error Estimation for sin(0.5)

a_4 = 0.000096 gives |E_3| ≤ 0.000096.

Convergence of Series

Assured by the Alternating Series Test.

Absolute Convergence

An alternating series converges if the series of absolute values also converges.

Conditional Convergence

If an alternating series converges while the series of absolute values diverges.

Example of Conditional Convergence

The alternating harmonic series: ∑ (-1)^(n+1)/n converges, but ∑ 1/n diverges.

Relation to Taylor Series

The Alternating Series Error Bound Theorem is often applied in Taylor series expansions.

Graphical Interpretation

Visualizing series and their convergence through plots of partial sums.

Error Analysis in Numerical Methods

Assessing reliability of numerical solutions using the theorem's error bounds.

Choosing the Number of Terms

Determined by the desired accuracy in approximating series.

Key Points Summary

The theorem provides error bounds, conditions for convergence, and applications in various fields.