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.

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

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}| ).

For a convergent alternating series, the error in approximating the sum by the nth partial sum is no greater than the absolute value of the (n+1)th term.

It's used to estimate the accuracy of approximating the sum of a convergent alternating series with a finite number of terms.

The series must be alternating, and the absolute value of the terms must be decreasing and approaching zero.

It provides a practical way to determine the accuracy of numerical approximations of alternating series.

It relies on the convergence of the alternating series to provide a meaningful error bound.

It allows us to determine how many terms are needed to achieve a desired level of accuracy in approximating the sum of an alternating series.

If ( |a_{n+1}| \geq |a_{n+2}| ) and ( \lim_{n \to \infty} a_n = 0 ), then ( |s - s_n| \leq |a_{n+1}| ).

By setting up the inequality ( s_n - |a_{n+1}| \leq s \leq s_n + |a_{n+1}| ).

It ensures that the terms are decreasing and approaching zero, which is necessary for the series to converge and for the error bound to be valid.

It provides a way to quantify the error when approximating an infinite sum with a finite partial sum, allowing for controlled accuracy.