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

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Answer 1 · 2018-08-14T16:03:43

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.