does the series from n=1 to infinity of sqrt { ln n. e^-sqrt(n) }, converge ? and if so, is the sum less than infinity.

Notice that
log
(√
e−
√
n log n
)
= −1
2
√
n+
1
2
log log n ≤ −2 log n
for sufficiently large n. The last inequality follows since 2 logn + 12 log log n ≤
1
2
√
n for large n, as the square root function grows much quicker than the
logarithm. This tells us that √
e−
√
n logn ≤...