Use divergent test to determine convergence of the following series?

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

1 Answer
Mar 25, 2018

Diverges.

Explanation:

The Divergence Test tells us that to determine the divergence of a series suma_n, take lim_(n->oo)a_n.

If lim_(a->oo)a_n ne 0, the series suma_n diverges.

The converse is NOT true; if the limit is zero, the series may or may not converge.

Here, a_n=n/sqrt(2n^2+1). Take the limit:

lim_(n->oo)n/sqrt(2n^2+1)=lim_(n->oo)(n/n)/sqrt((2n^2+1)/n^2)=lim_(n->oo)1/sqrt(2+1/n^2)=1/sqrt(2) ne 0

Thus, the series diverges.