Hey Balarka, Daniel is upgrading the forum software to a completely new platform. It should be back up soon.
I agree that fun shouldn't introduce wrong concepts. PS: Yes, I'll add one when I start the next post.
Ok, then you are looking at rigor but as I said these tutorials will be for fun. I want a high school student to understand it. It would be great to have your comments when I start posting bout that, though. I know I can learn from you as I am haven't read that much about these concepts.
What is your definition of Regularization ?
We can extend the zeta function to analytic function in the whole complex plane except at 1. By definition we have $$\zeta(s) =\sum \frac{1}{n^s}$$ So zeta on the left is analytic for all $s \neq 1$ but the sum on the right does only converge for $Re(s) >1$. So the analytic continuation of zeta has enabled us to give values to the divergent sum on the right.
Hey Balarka , the idea of regularization is giving a finite value for divergent series or integrals. Consider the following $$\zeta(-2)= 0 $$ $$1+4+9+\cdots = 0 $$
Okay, thanks. I think the former one is just what I wanted.
Hey Balarka, I do not know of a closed for that sum. But there are some identities related to this problem: $$ \begin{align*} \sum_{k=1}^p \frac{H_k^{(2)}}{k}+\sum_{k=1}^p \frac{H_k}{k^2} &= H_p^{(3)}+H_p^{(2)}H_p \\ \sum_{k=1}^p \frac{H_k^{(2)}}{k}+\sum_{k=1}^p \frac{(H_k)^2}{k} &= \frac{(H_p)^3+3H_p H_p^{(2)}+2H_p^{(3)}}{3} \end{align*} $$ To prove these, we can use induction.
Possibly, that's my name there. I don't gravitate toward your usual tags, I suppose. I'm more in multivariable calculus, calculus and lower level ones.
Good question , where did you get that ? I never tried something similar but surely I will give it a shot !
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