`sum_(n=1)^oo n^2/(n^2+1)` Verify that the infinite series diverges

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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

To verify if the series diverges, apply the nth-Term Test for Divergence.

It states that if the limit of `a_n` is not zero, or does not exist, then the sum diverges.

`lim_(n->oo) a_n != 0`      or      `lim_(n->oo) = DNE`

`:.` `sum` `a_n` diverges

Applying this, the limit of the term of the series as n approaches infinity is:

`lim_(n->oo) a_n`

`=lim_(n->oo) n^2/(n^2+1)`

`= lim_(n->oo) n^2/(n^2(1+1/n^2))`




The limit of the series is not zero. Therefore, by the nth-Term Test for Divergence, the series diverges.

Approved by eNotes Editorial Team