Alternating Series Error Bound Theorem

Alternating Series

Alternating Series

A series where the terms alternate in sign, e.g., ∑n=1∞(−1)n−1an.

Alternating Series Test

A criterion used to determine the convergence of an alternating series.

Convergence Conditions

An alternating series converges if its terms are monotonically decreasing and their limit approaches zero.

Monotonically Decreasing

A sequence where each term is less than or equal to the previous term.

Partial Sum

The sum of the first N terms of a series, denoted as SN.

Error in Approximation

The difference between the actual sum (S) and the partial sum (SN), expressed as |S - SN|.

Alternating Series Error Bound Theorem

States that the error in approximating the sum of an alternating series is less than or equal to the absolute value of the first omitted term, |S - SN| ≤ aN+1.

Absolute Error Bound

An error bound that does not consider the sign of the terms in the sequence.

Omitted Term

The first term that is not included in a given partial sum.

aN+1

The term in the sequence used to estimate the error in an alternating series.

Example of Alternating Harmonic Series

S = ∑n=1∞(−1)n−1n, which converges to the alternating harmonic series.

Error Bound Calculation Example

For S3 in the alternating harmonic series, |S - S3| ≤ a4 = 1/4.

Approximation Accuracy

The degree to which an approximation is close to the actual value.

Numerical Analysis

A field of study that uses algorithms to approximate the solutions to mathematical problems.

Transcendental Functions

Functions that are not algebraic, such as ln(1+x) and e^x.

Engineering Applications

In fields like engineering and physics, alternating series and their error bounds help in calculating precise results.

Taylor Series

A series expansion that expresses a function as an infinite sum of terms based on its derivatives.

Connection to AP Calculus

The theorem is crucial for understanding series and is often featured on the AP exam.

Conceptual Understanding

The ability to grasp the principles and relationships underlying mathematical concepts.

Leibniz Criterion

Another name for the Alternating Series Test; helps test for the convergence of an alternating series.

Example of Smaller Terms Series

For an = 1/n^2, the series converges, and the error bound can be calculated similarly.

Applications in Calculus

The theorem assists in estimating sums of series, enhancing students' calculus skillset.

∞ (Infinity) in Series

Indicates that the sum extends indefinitely beyond finite bounds.

Importance of Error Bounds

They provide a way to quantify how close an approximation is to the actual sum.

Harmonic Series

A specific type of alternating series, where terms are inverses of positive integers.

First Omitted Term's Role

The size of the first omitted term directly influences the accuracy of the approximation.