Convergence/divergence of a series

[miglo]

Homework Statement

the original question asks for the radius and interval or convergence, and the values of x for which the series $$\sum_{n=1}^{\infty}(1+\frac{1}{n})^{n}x^n$$ will converge absolutely and conditionally
ive figured out the radius of convergence (R=1) and my interval of convergence (-1<x<1) but now i need to check the endpoints of the interval of convergence to see where it converges absolutely and conditionally if it does in fact converge conditionally
so I am trying to show whether $$\sum_{n=1}^{\infty}(-1)^{n}(1+\frac{1}{n})^{n}$$ converges or diverges

Homework Equations

absolute convergence test
alternating series test

The Attempt at a Solution

so i first tried using the absolute convergence test which meant that i only had to look at the series without the (-1)^n, and then i applied the divergence test or n-th term test, taking the limit of the sequence as n approached infinity but that just gives me e^1 which is greater than 0 therefore it diverges but this doesn't imply that the original series with the (-1)^n attached to it will diverge so i have to use the alternating series test
but the sequence $(1+\frac{1}{n})^{n}$ isn't decreasing nor will the limit as n approaches infinity go to 0, so i can't use the alternating series test to show whether its converging or diverging
so what do i do now? can i just say since it doesn't satisfy the requirements necessary to make it convergent by the alternating series test then it will diverge? or is there some other method I am not seeing?

 

[lurflurf]

All that stuff does not matter. A series can only converge if lim a_n=0, so this series cannot converge.

 

[miglo]

i thought it would something really simple, i guess i was just over thinking it
thanks!