Maria Agnesi

I already have proven that the following series converges.

In this article, I will show that:

Lemma 1. The following equations hold:

Proof. Define s_n as

then a simple calculation shows that:

Dividing both sides by 2sin(x/2), the conclusion follows. By the same manner, define t_n as

then we have

Dividing both sides by 2sin(x/2), the conclusion follows.

Lemma 2. The following sequence converges on any closed subntervals of (0, 2π), while α is a positive real number.

Proof. Define a_n(x) and b_n(x) as following:

Then we have

So, ∑a_n is bounded. Moreover, b_n decreases and converges to 0. Thus, by Dirichlet's test, ∑a_nb_n converges uniformly.

Lemma 3. The following equation holds:

Proof. For the series converges uniformly, we have:

Thus the conclusion follows.

If we substitute x by 1, we have the desired equation:

Posted by Maria Agnesi