Alternating Series Error Bound

An alternating series flips sign every term and, when it converges by the alternating series test, its terms shrink to zero. Picture the partial sums: S1 is too big, S2 pulls back and is too small, S3 overshoots again, each time by a smaller amount. Because the terms keep shrinking, this zigzag squeezes tighter and tighter around the true sum L. That squeezing is why the leftover error after stopping at S_n can never exceed the size of the very next term, a_(n+1): that term is literally the distance you jump on the next step, and the true value sits inside that jump.

The mechanism is just the sandwich the zigzag creates. S_n and S_(n+1) sit on opposite sides of L, so L is trapped between them. The gap between S_n and S_(n+1) is exactly a_(n+1), so the error |L - S_n| is at most a_(n+1). No ratio tests, no integrals, no comparison work needed, which is what makes this bound so fast to use once the series qualifies.

This sits next to the broader toolkit of error estimates in the series unit, alongside the remainder estimate for integral test series and Taylor series remainder bounds. It is the cheapest of the three: one term, no calculus. But it only applies when the series truly alternates every term and the terms decrease to zero; otherwise you need a different bound, like the Lagrange error bound for a Taylor polynomial.