`a_n = ln(n^3)/(2n)` Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges, find its limit.

Expert Answers

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

`a_n=(ln(n^3))/(2n)`

The first few terms of the sequence are:

`0` ,  `0.5199` ,  `0.5493` ,  `0.5199` ,  `0.4828` ,  `0.4479` ,  `0.4170` ,...

To determine if the sequence converge as the n becomes larger, take the limit of the nth-term as n approaches infinity.

`lim_(n->oo)a_n`

 `=lim_(n->oo) (ln(n^3))/(2n)`

To take the limit of this, apply  L'Hospital's Rule.

`=lim_(n->oo) ((ln(n^3))')/((2n)')`

`=lim_(n->oo) (1/n^3*3n^2)/2`

`=lim_(n->oo) (3/n)/2`

`=lim_(n->oo) 3/(2n)`

`= 3/2 lim_(n->oo) 1/n`

`=3/2*0`

`=0`

Therefore, the sequence is convergent.  And the terms converges to a value of 0. 

Approved by eNotes Editorial Team