Just to clarify all the numbers in this post are going to exist in . Given a sequence where we have a collection of partial sums indexed by and defined by . If the sequence converges to (In the usual – way) we say the series converges and write . For completness if the sequence diverges, the series is said to diverge.
If you know sequences really well, you know series really well as every theorem about sequences can be stated in terms of series (putting and for ). In particular the monotone convergence theorem has an instantaneous counterpart for series.
Theorem: A series of non-negative real numbered terms converges if and only if the partial sums form a bounded sequence.
I’m going to omit the proof here but it is a quick application of the monotone convergence theorem to the partial sums. So why bring this up? Well if we impose that the terms in our series are decreasing monotically (which can appear in applications) we can apply the following theorem of Cauchy. What is interesting about this theorem is that a ‘thin’ subsequence of determines the convergence or divergence of the series.
Theorem: Suppose are real numbers. Then the series converges if and only if the series converges.
Proof: By the previous theorem it suffices to consider only the boundedness of the partial sums. Let us write and . We will look at two cases, when and when .
For we have where the first inequality followed from and the second inequality from the hypothesis.
When we have where the first inequality follows from and the second (you guessed it) follows from our hypothesis.
Bringing these together we conclude that the sequence and are either BOTH bounded or BOTH unbounded which completes the proof.
When I came across this I thought it was pretty astounding (hence why it has made it onto the blog) so lets see it it action. We will use it to deduce for that converges if and diverges if .
The monotonicity of the logarithmic function implies increases which puts us in good position to apply our theorem. This leads us to the following which is enough as a proof.