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.
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
So, the alternating harmonic series converges.