/sci/ - Science & Math

>>15924439
Consider the following formula.
[eqn]
\sum_{n=1}^\infty \left(
\left( \frac{r}{1+r^2} \right)^{2n-1} \sum_{j=1}^n \frac{(2n-1)!}{(2n-1-j)! j! b^{(2n+1-2j)/2}}
-\left( \frac{r}{1+r^2} \right)^{2n} \sum_{k=1}^n \frac{(2n)!}{(2n-k)! k! b^{(2n+2-2k)/2}}
\right)
[/eqn]
with [math] r = \frac{-1}{\sqrt{3}} [/math] and [math] b = 3 [/math]. As you can see from the included image, my plot on Desmos strongly suggests that this infinite sum adds up to [math] \frac{-11}{9} [/math]. However, a plot is not a proof. Can anyone please prove (or disprove) my hunch?