Conditional Convergence
Conditional convergence is really the difference between convergence and absolute convergence.
Suppose we have a series
which converges. If
diverges then the series is said to converge conditionally.
Example:
converges conditionally, since
the well known harmonic series diverges but
Now we use the comparison test:
The last expression above is a basic convergent series so
converges.
It is obvious that every absolutely convergent series is convergent.
What follows is a very useful theorem on conditional convergence.
Theorem (Alternating Series Test)
Suppose
is a sequence of real numbers such that
and thatconverges to zero. Then
converges.
Proof: Set
withthen the series has partial sum
Then
Since
we have
Also
hence the even partial sums and the odd partial sums are bounded, and monotonically increasing and decreasing respectively. Since
andconverges to zero, the theorem is proved.
Example:
converges by the alternating series test since
as