use ratio test to solve

Evaluate `sum_(n=1)^(oo)(ln(n))/n^3 ` We are asked to use the ratio test to determine if the series converges.

We take the limit as n tends to infinity of the (n+1)st term over the nth term: if the limit is less than 1 in absolute value the series converges, if greater than 1 the series diverges, and if equal to 1 there is no conclusion.

`lim_(n->oo)(ln(n+1)/(n+1)^3)/(ln(n)/n^3) `

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

`= lim_(n->oo) (ln(n+1))/(ln(n))*lim_(n->oo)n^3/(n+1)^3 ` Using L'Hopitals rule on the first factor

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

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

Thus the ratio test is inconclusive.

(The series does converge -- use a comparison test. Note that `(ln(n))/n^3<n/n^3=1/n^2 `
which converges.)

Comments

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