When I’ve followed a notes that show how obtain a similar asymptotic using Abel summation formula, my case with $a_n=\chi(n)$, the characteristic function taking the value 1 if $p$ is prime (in a twin prime-pair, thus caution I’ve defined $\chi(p+2)$ as zero) and $f(x)=x^{\alpha}$, which $\alpha>-1$, and Prime Number Theorem, in my case I am […]

Does Abel or Cesaro summable imply Borel summable for a series? In other words, for a sequence $(a_n)$ and its partial sums $(s_n)$, is it true that: $\lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n-1} s_k = A \Longrightarrow \lim_{t \to \infty}e^{-t}\sum_{n=0}^{\infty}s_n\frac{t^n}{n!} = A$ $\lim_{x \to 1^-}\sum_{n=0}^{\infty}a_nx^n = A \Longrightarrow \lim_{t \to \infty}e^{-t}\sum_{n=0}^{\infty}s_n\frac{t^n}{n!} = A$. Is there a proof of […]

Repeated union is written as: $$\bigcup_{i=0}^na_i$$ Repeated logical conjunction is: $$\bigwedge_{i=0}^na_i$$ Etc. So why isn’t repeated addition: $$\operatorname{\huge+}\limits_{i=0}^n{}^{\Large a_i}$$ Why use Sigma and Pi for sums and products? Everything else is just a bigger version of the symbol.

Suppose I had a function that satisfied the property $f(x)=f(x-1)+g(x)$. For any $x\in\mathbb N$, it is easy enough to see that this boils down to the statement $$f(x)=f(0)+\sum_{k=1}^xg(k)$$ If we return to our functional equation and differentiate it $n$ times, we get $$f^{(n)}(x)=f^{(n)}(x-1)+g^{(n)}(x)$$ Once again, for any $x\in\mathbb N$, we have $$f^{(n)}(x)=f^{(n)}(0)+\sum_{k=1}^xg^{(n)}(k)$$ One could then […]

Find the value of $$S=\sum_{n=1}^{\infty}\left(\frac{2}{n}-\frac{4}{2n+1}\right)$$ My Try:we have $$S=2\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{2}{2n+1}\right)$$ $$S=2\left(1-\frac{2}{3}+\frac{1}{2}-\frac{2}{5}+\frac{1}{3}-\frac{2}{7}+\cdots\right)$$ so $$S=2\left(1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\cdots\right)$$ But we know $$\ln2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$$ So $$S=2(2-\ln 2)$$ Is this correct?

Regarding the proof by Tony Padilla and Ed Copeland that $1+2+3+\dotsb=-\frac{1}{12}$ popularized by a recent Numberphile video, most seem to agree that the proof is incorrect, or at least, is not sufficiently rigorous. Can the proof be repaired to become rigorously justifiable? If the proof is wrong, why does the result it computes agree with […]

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