AP Calculus BC glossary
Alternating harmonic series
The alternating harmonic series converges to the natural logarithm of two, even though the harmonic series itself diverges. It is the standard example of conditional convergence: it converges as written but not once you take absolute values.
It converges by the alternating series test, since the terms decrease in size to zero. Absolute values turn it back into the harmonic series, which diverges, so the convergence is conditional.
The mistake
Assuming alternating signs guarantee convergence. The terms must also decrease in magnitude to zero, and both conditions have to be stated.
Appears in: Unit 10: Infinite Sequences and Series (BC)