A series where the terms alternate in sign.

The maximum possible difference between the true sum of the series and a partial sum approximation.

The sum of a finite number of terms of a series.

The term immediately following the last term included in the partial sum.

It represents the maximum possible error when approximating the infinite sum by a partial sum.

An alternating series that approaches a finite limit as the number of terms increases indefinitely.

It refers to the alternating signs of the terms, which allows for a simple error bound calculation.

To estimate the accuracy of approximating an infinite series with a finite partial sum.

Generally, the more terms used, the smaller the error bound, leading to a more accurate approximation.

\( a_i ) represents the absolute value of the first omitted term, which serves as the error bound.

$|s - s_{i-1}| leq a_i$

The true sum of the infinite alternating series.

The partial sum of the first ( i-1 ) terms of the series.

The absolute value of the ( i )-th term of the series (the first omitted term).

The absolute difference between the true sum ( s ) and the partial sum ( s_{i-1} ) is less than or equal to ( a_i ).

By rewriting the inequality as ( s_{i-1} - a_i leq s leq s_{i-1} + a_i ).

If the alternating series is given by $\sum_{n=1}^\infty (-1)^n a_n$, then the nth term is $(-1)^n a_n$.

Sum the first ( n ) terms of the series: ( s_n = \sum_{i=1}^n (-1)^i a_i ).

$\sum_{n=1}^\infty (-1)^n a_n$ or $\sum_{n=1}^\infty (-1)^{n+1} a_n$, where ( a_n > 0 ) for all ( n ).

The error bound is given by the absolute value of the (n+1)-th term, i.e., ( |a_{n+1}| ).

We use a partial sum to approximate the infinite sum. The Alternating Series Error Bound Theorem helps quantify the accuracy of this approximation.

It provides a simple way to determine the maximum possible error when using a partial sum to approximate the sum of a convergent alternating series.

An accurate estimation means that the difference between the estimated value (partial sum) and the true value (infinite sum) is small, as quantified by the error bound.

Because the alternating signs cause the partial sums to oscillate around the true sum, with each term bringing the partial sum closer.

Smaller terms lead to a smaller error bound, indicating a more accurate approximation with fewer terms.

Generally, using more terms in the partial sum leads to a smaller error bound and a more accurate estimation of the infinite sum.

The alternating signs ensure that the error is no larger than the absolute value of the first omitted term.

The series must be convergent for the error bound to provide a meaningful estimate of the accuracy of the partial sum.

The oscillating nature of the alternating series causes the partial sums to converge towards the true sum, with each term correcting the previous partial sum by at most the absolute value of that term.

The error bound provides a way to quantify how close a partial sum is to the limit (true sum) of the infinite alternating series.