Converges Or Diverges: $\sum _{n=1}^{\infty }\:e^{-\sqrt{n}}$ Converges Or Diverges: 
Attempt:  
int_1^∞ e^(-sqrt(n)) dn 
t = -sqrt(n); dt = -dn/(2*sqrt(n)); 
int -2*sqrt(n)/[-2*sqrt(n)] * e^(-sqrt(n)) dn 
int (2t)*(e^t) dt 
u = 2t; du = 2 dt; dv = e^t dt; v = e^t 
(2t)*(e^t) - int 2*e^t 
2t*(e^t) - 2*(e^t) 
2 * (e^t) * (t - 1) 
2 * e^(-sqrt(n)) * (-sqrt(n) - 1) 
-2 * e^(-sqrt(n)) * (sqrt(n) + 1) 
limit n->∞ [-2 * e^(-sqrt(n)) * (sqrt(n) + 1)] 
[-2 * d/dn (sqrt(n) + 1)] / [d/dn e^sqrt(n)] 
limit n->∞ -2*e^(-sqrt(n)) = 0 

Converges by comparison with $~\displaystyle\sum_{n=1}^\infty e^{-2\ln n}~=~\sum_{n=1}^\infty\frac1{n^2}$
