Interactive Real Analysis
- part of Alternating Series Test
A series of the form
with
b n 
0
is called Alternating Series. If the sequence
is decreasing and converges to zero, then the sum converges.
Context
This test does not prove absolute convergence. In fact, when checking
for absolute convergence the term 'alternating series' is meaningless.
with
b n 0
is called Alternating Series. If the sequence
is decreasing and converges to zero, then the sum converges.
ContextIt is important that the series truly alternates, that is each positive term is followed by a negative one, and visa versa. If that is not the case, the alternating series test does not apply (while Abel's Test may still work).
| Examples 4.2.20: | |
| |
converge or diverge ?
?
Let a n = (-1) n. Then the formal sum
has bounded partial sums (although the sum does not itself converge.
Why not ?). Then, with the given choice of
Abel's test applies directly, showing that the series indeed converges.
