Primitive root mod p

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Infinite products 28 February, 2008

Filed under: Mathematics — Nikolas Karalis @ 12:23 am

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 LaTeX.
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 LaTeX 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 a_n be a sequence of positive numbers. Then the infinite product prod_{n=1}^{infty} {(1+a_n)} converges if and only if the series sum_{n=1}^{infty} a_n converges.

It is easy to see that we can use this result also for a sequence of negative numbers, when -1 leqslant a_n leqslant 0 .

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

sum_{k=1}^{infty} frac{(-1)^{k+1}}{k}=1-frac{1}{2}+frac{1}{3}-frac{1}{4}+...

We take prod_{n=1}^{infty} {(1+a_n)} = (1+1)(1-frac{1}{2})(1+frac{1}{3})(1-frac{1}{4}) = 2 times frac{1}{2} times frac{4}{3} times frac{5}{4} times ... = 1

So, the alternating harmonic series converges.

QED

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