Theorem:
The Alternating Harmonic Series
The series
is called the Alternating Harmonic series. It converges but
not absolutely, i.e. it converges conditionally.
Proof:
There are many proofs of this fact. For example. the series of
absolute values is a p-series with p = 1, and
diverges by the p-series test. The original series
converges, because it is an alternating series, and the
alternating series test applies easily. However, here is a
more elementary proof of the convergence of the alternating
harmonic series.
We already know that the series of absolute values does not
converge by a previous example. Hence, the series does not
converge absolutely. As for regular convergence, consider the
following two partial sums:
We have that
- S 2n+2 - S 2n =
1 / (2n+1) - 1 / (2n+2) > 0
- S 2n+3 - S 2n+1 =
- 1 / (2n+2) + 1/ (2n+3) < 0
which means for the two subsequences
- { S 2n } is monotone increasing
- { S 2n+1 } is monotone decreasing
For each sequence we can combine pairs to see that
- S 2n
1 for all n
- S 2n+1
0 for all n
Hence, both subsequences are monotone and bounded, and must therefore be convergent.
Define their limits as
- lim S 2n = L and
lim S 2n+1 = M
Then
- | M - L | =
| lim (S 2n+1 - S 2n) | = 1 / (2n+1)
which converges to zero. Therefore, M = L, i.e. both
subsequences converge to the same limit. But this common limit
is the same as the limit of the full sequence, because: given
any 
> 0 we have
- there exists an integer N such that
| L - S 2n | <
if n > N
- there exists an integer M such that
| L - S 2n+1 | <
if n > M
Now set K = max(N, M). Then, for the above
> 0 we have
- | L - S n | <
for n > K
because n is either even or odd. Hence, the alternating harmonic
series converges conditionally.
To Theory |
Glossary |
Map
(bgw)
