Background song : Molotov – Frijolero

Since when this blog migrated from my own server to the WordPress one, I always disliked some of the restrictions this implied. One of the most important ones was the lack of .

But as you may have already guessed from the previous sentence I have great news.

Wordpress has a new feature, so we can write normal code in our posts and comments.

So, I inaugurate this new feature, by extending a result I found at The Everything Seminar, a blog I found recently and since then reading it.

At the post Convergence of Infinite, we see a simple but strong convergence test.

** Theorem.**

Let be a sequence of positive numbers. Then the infinite product converges if and only if the series converges.

It is easy to see that we can use this result also for a sequence of negative numbers, when .

Using this we can derive the classic result of the divergence of the harmonic series.

We can also show that the alternating harmonic series converges.

**Alternating Harmonic Series**

We take

So, the alternating harmonic series converges.

**QED**