Question

Use the limit comparison test to determine whether ∑n=7[infinity]an=∑n=7[infinity]9n3−6n2+76+3n4 converges or diverges. (a) Choose a series ∑n=7[infinity]bn with terms of the form bn=1np and apply the limit comparison test. Write your answer as a fully simplified fraction. For n≥7, limn→[infinity]anbn=limn→[infinity] (b) Evaluate the limit in the previous part. Enter [infinity] as infinity and −[infinity] as -infinity. If the limit does not exist, enter DNE. limn→[infinity]anbn =

Answer 1

It does not converge.

Explanation:

Since

`6n^2 \leq 9n^3`

and

`0 \leq 9n^3 - 6n^2`

Adding 7 on both sides of the inequalty we get

`7 \leq 9n^3 - 6n^2 +7`

and we know that the sum
`\sum_(n=0)^(\infty) 7` diverges. Therfore by the comparison test

`\sum_(n=0)^(\infty) 9n^3 - 6n^2 +7`

Does not converge.