An alternating series is a series where sequential terms alternate signs. This is typically represented by a (-1) to some power multiplied by a sequence of positive terms (a_n). The video provides examples of alternating series, including the alternating harmonic series.

The Alternating Series Test has two main conditions for convergence: 1) The limit of the sequence of positive terms (a_n) as n approaches infinity must be zero. 2) The sequence of positive terms (a_n) must be decreasing for all n. If both conditions are met, the alternating series converges.

The video provides a graphical explanation for why an alternating series converges if its terms are decreasing and tend to zero. The partial sums 'ping-pong' back and forth, getting progressively closer to a single sum due to the decreasing nature of the terms.

The test is applied to the alternating harmonic series (Σ (-1)^(n-1) * 1/n). It is shown that the limit of 1/n as n approaches infinity is 0, and 1/n is a decreasing sequence. Therefore, the alternating harmonic series converges, contrasting with the divergent harmonic series.

Another example (Σ (-1)^(n+1) * 2n/(4n-1)) is analyzed. Even though the sequence of positive terms is decreasing, the limit of 2n/(4n-1) as n approaches infinity is 1/2, not 0. Therefore, by the Divergence Test (applied to the a_n part), this series diverges.

A third example (Σ (-1)^n * (n+1)/(√(n^2-1))) is examined. The limit of (n+1)/(√(n^2-1)) as n approaches infinity is 0. To show it's decreasing, a derivative of the corresponding continuous function is used, confirming the sequence is indeed decreasing. Thus, the series converges by the Alternating Series Test.

When a series converges to a sum (S) and we only calculate a partial sum (S_n), there is an error (R_n). This error is the difference between the actual sum and the partial sum (R_n = S - S_n). The concept of R_n is the remainder or the 'rest' of the sum after 'n' terms.

For alternating series, the absolute value of the error (R_n) is always less than or equal to the absolute value of the first neglected term, a_(n+1). This unique property of alternating series allows for a simple way to bound the error.

An example (Σ (-1)^n * 1/n!) is used to demonstrate how to determine the number of terms needed to achieve a specific error bound. By setting the absolute value of the (n+1)-th term (a_(n+1)) less than the desired error, one can solve for 'n' to find the required number of terms for the partial sum.

The process involves finding 'n' such that |a_(n+1)| is less than the specified error (e.g., 0.0005). Once 'n' is found, the sum of the first 'n' terms (S_n) is calculated. This partial sum will be within the given error tolerance of the true sum of the series.