sum_ n=1^infinity n/(sqrt(n^2+1))

Question

verify that th infinite series diverges

Answer 1

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→∞.