convergence of $\sqrt{-e-e²+e^3+\frac{1}{-e-e²+e^3}-\frac{1}{\sqrt{-e-e²+e^3}}+\frac{1}{\sqrt[3]{-e-e²+e^3}}-\frac{1}{\sqrt[5]{-e-e²+e^3}}+\cdots }$?

Question

I want to check convergence of sequence which is related to exponential function i have come up with this sequence such that I run some computations in wolfram alpha it's seems that is convergent , but my Goal is to know if the bellow sum could be an integer number .

$S=\sqrt{-e-e²+e^3+\frac{1}{-e-e²+e^3}-\frac{1}{\sqrt{-e-e²+e^3}}+\frac{1}{\sqrt[3]{-e-e²+e^3}}-\frac{1}{\sqrt[5]{-e-e²+e^3}}+\cdots }$

?

Answer 1

I guess you want to write the sequence defined by

$$S^2:=C+\sum_{k\ge 1}(-1)^{k+1} C^{-1/k}$$

for $C:=-e-e^2+e^3$. The series doesnt converge because the inner sequence $(C^{-1/k})_{k\in\Bbb N_{>0}}$ doesnt converge to zero.