Help determining convergence of series

[ag_roque]

This is what I've got.
I need to demonstrate (absolute) convergence but ratio test is inconclusive.
Any help is greatly appreciated.
∞
∑ e^((-n)(1+i))
n=1

 

[pka]

I need to demonstrate (absolute) convergence but ratio test is inconclusive.
Any help is greatly appreciated.
∞
∑ e^((-n)(1+i))
n=1

Please review what you have posted. Is it exactly as written?
I wonder if you really mean the complex \(\bf{\mathcal i}\)?

 

[ag_roque]

Please review what you have posted. Is it exactly as written?
I wonder if you really mean the complex \(\bf{\mathcal i}\)?

YES
I MEANT THE COMPLEX i

 

[ag_roque]

Inferior limit n=1, superior limit infinity.

Given: [MATH] ∑(e)^{-n(i+1)}[/MATH]

 

[pka]

Inferior limit n=1, superior limit infinity.

Given: [MATH] ∑(e)^{-n(i+1)}[/MATH]

Thank you for a much clearer representation.
Now it is clearly a geometric convergent series. SEE HERE

 

[ag_roque]

not so clear for me.
I still dont know how to make the demonstration

 

[Cubist]

Have you tried splitting [math] \mathrm{e}^{-n \left(?+1\right)} [/math] into real and imaginary parts?