Question

Use divergent test to determine convergence of the following series?

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

Answer 1

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.