Calculus II, Lecture 18: Alternating series

Lecture 18: Alternating series.

A series is alternating if the terms alternate in signs. This is usually due to the presence of (-1)^n=1,-1,1,-1,... in the sum. We begin with the Alternating Test, which gives a criterion for convergence. The related Alternating Remainder Theorem tells us how quickly a series converges by bounding the difference in the limit and the n'th partial sums below the (n+1)'th term. Finally, compare alternating series that converge absolutely and those that converge conditionally, which depends on whether the series of absolutely values |a_n| converges or diverges. We uncover a remarkable, almost paradoxical, fact that when a series converges conditionally, we can re-arrange its terms to get a series that converges to anything we want!