Articles of summation method

Sum of Gaussian Sequence

I am looking for a closed form or an estimation for the sum of a Gaussian sequence expressed as $$ \sum_{x=0}^{N-1} e^{\frac{-a}{N^2} \: x^2} $$ where $a$ is a constant positive integer. The interesting part is that I have simulated this sequence using MATLAB for $N=0$ to $10^6$ and found that the result is linear […]

A conditional asymptotic for $\sum_{\text{$p,p+2$ twin primes}}p^{\alpha}$, when $\alpha>-1$

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 […]

Abel/Cesaro summable implies Borel summable?

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 […]

Why sigma notation?

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.

A method for evaluating sums/discrete functions by assuming they can be made continuous and differentiable?

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 $\sum_{n=1}^{\infty} \frac{2}{n}-\frac{4}{2n+1}$

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?

Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?

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 […]