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## Main Question or Discussion Point

Along with some friends, i was trying to calculate the end product of a series(x^(1-x)). I for a finite value, i got a number close to arcsinh(SQRT 6), and by comparing the functions i believe it converges at infinity. However, i am unable to 100% prove this, and i am asking if the summation is correct in particular and whether a similar series is already in existence.

..[itex]\infty[/itex]

[itex]\sum[/itex][itex]\chi^{1-\chi}=arcsinh(\sqrt{6})[/itex]

..[itex]\chi=1[/itex]

(PS does anyone know how to type summations much faster than using the sigma notation, then typing below and above etc. I am not too familiar with Latex

Another summation that i would like to confirm:

..[itex]\infty[/itex]

[itex]\sum[/itex][itex]\frac{1}{\Gamma\chi}=e[/itex]

..[itex]\chi=1[/itex]

This can be derived by representing e^x with a summation.

..[itex]\infty[/itex]

[itex]\sum[/itex][itex]\chi^{1-\chi}=arcsinh(\sqrt{6})[/itex]

..[itex]\chi=1[/itex]

(PS does anyone know how to type summations much faster than using the sigma notation, then typing below and above etc. I am not too familiar with Latex

Another summation that i would like to confirm:

..[itex]\infty[/itex]

[itex]\sum[/itex][itex]\frac{1}{\Gamma\chi}=e[/itex]

..[itex]\chi=1[/itex]

This can be derived by representing e^x with a summation.