Conditional convergence is really the difference between convergence and absolute convergence.
Suppose we have a serieswhich converges. Ifdiverges then the series is said to converge conditionally.
Example:converges conditionally, sincethe well known harmonic series diverges but
Now we use the comparison test:
The last expression above is a basic convergent series soconverges.
It is obvious that every absolutely convergent series is convergent.
What follows is a very useful theorem on conditional convergence.
Theorem (Alternating Series Test)
Supposeis a sequence of real numbers such thatand thatconverges to zero. Thenconverges.
Proof: Setwiththen the series has partial sum
Sincewe haveAlsohence the even partial sums and the odd partial sums are bounded, and monotonically increasing and decreasing respectively. Sinceandconverges to zero, the theorem is proved.
Example:converges by the alternating series test since as