Question

How do you use the ratio test to test the convergence of the series `∑k/(3+k^2)` from k=1 to infinity?

Answer 1

Let:

`a_k = k/(3+k^2)`

and evaluate the ratio:

`abs (a_(k+1)/a_k) = abs ( ( (k+1)/(3+(k+1)^2) ) / ( k/(3+k^2) ))`

`abs (a_(k+1)/a_k) = ( (k+1)/k ) ( (3+k^2) /(4+2k+k^2))`

We have that:

`lim_(k->oo) abs (a_(k+1)/a_k) = 1`

so the ration test is in effect inconclusive to determine whether the series:

`sum_(k=1)^oo a_k`

is convergent.

However if we consider the harmonic series:

`sum_(k=1)^oo 1/k`

which is divergent and we apply the limit comparison test, we can see that:

`lim_(k->oo) a_k/(1/k) = lim_(k->oo) k^2/(3+k^2) = 1`

so, as the limit is finite, the two series have the same character and we can conclude that;

`sum_(k=1)^oo a_k`

is divergent.