n/√(n²+1)=
n/(n√(1+1/n²)=
1/√(1+1/n²).
If we write the first few terms of the series we get:
1/√2, 2/√5, 3/√10, 4/√17, ...
As n gets larger the terms get closer to 1, so the sum of the series grows by almost 1 with each successive term, and therefore must diverge as n→∞.