Answer by robjohn for Limit approach to finding $1+2+3+4+\ldots$
Explanation for the Behavior of the Series (Parts $2$ and $3$) Note that $$ \begin{align} \sum_{k=0}^\infty e^{ikx} &=\frac1{1-e^{ix}}\\ &=\frac12+\frac i2\cot\left(\frac x2\right)\\...
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Answer by Nikolaj-K for Limit approach to finding $1+2+3+4+\ldots$
The infamous $-\frac{1}{12}$ pops up as the coefficient of of a range of expansions, such as $\dfrac{z}{{\mathrm e}^{-z}-1}=-1-\dfrac{1}{2}z-\dfrac{1}{12}z^2+{\mathcal O}(z^3)$ like in the Todd class...
View ArticleAnswer by Micah for Limit approach to finding $1+2+3+4+\ldots$
In Terence Tao's blog post about relating divergent series to negative zeta values, he uses Euler-Maclaurin summation to show that if $\eta$ is smooth and compactly-supported, and $\eta(0)=1$, then $$...
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Limit approach to finding $1+2+3+4+\ldots$
When exploring the divergent series consisting of the sum of all natural numbers $$\sum_{k=1}^\infty k=1+2+3+4+\ldots$$ I came across the following identity involving a one-sided limit:...
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