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Results tagged with divergent-series
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user 141375
1
vote
Theta-function in the lower half-plane
The answer to the question being asked for, on Generalized Analytical Continuation (GAC) of $\theta_i, i=1,2,3,4$ functions can be found in Addenda 2 of this MO discussion that extends Borel´s summati …
- 2,826
17
votes
Accepted
Value of divergent sum $\sum_{n=0}^\infty (-1)^n n^n$
The sum is equal to
$$
\int\limits_{0}^{\infty}\frac{\exp(-x)}{1+W_0(x)}\,\mathrm{d}x = 0.7041699604...
$$
where $W_0(x)$ is the Lambert-$W$ function.
Reference
Stephen Finch. "Errata and Addenda to M …
- 2,826
4
votes
A proposition for summing divergent series, but how should partial summation be defined at n...
Summation defined at non-natural values is also known as "fractional summation" (even if sum limits belong to ℂ). Markus Müller and Dierk Schleicher have provided a proper axiomatic framework defining …
- 2,826
13
votes
Accepted
Divergent series summation beyond natural boundaries
From those three examples, Rogers-Ramanujan's series belong to the class of basic hypergeometric series ($q$-series). It is a marginally logarithmic divergent series for $q > 1$ and it should be compu …
- 2,826