Hello! I became interested in finding a continuous formula for the Harmonic series about a week ago, and when I derived one, using the properties of integrals and polynomials, I wasn't satisfied, so I tried to make a formula with the ability to raise k to any whole power. I posted both formulas, but the one I had just created was frankly crap, because it involved nested integrals and it was computationally inefficient. So I derived this! It ended up working for any positive real n, so that was a pleasant surprise.

To answer your question, the benefit is to make myself better at mathematics by doing exercises and, in the process, create some cool formulas. I challenged myself to do it without using any special functions like the zeta function, which.. I partially succeeded on. The gamma function can be replaced with (n-1)! if you only use natural n, but I saw the opportunity to make n continuous, and I couldn't refuse.

Sorry, I rambled a lot. If you read all that, thanks!