Series/Sequence: divergent or convergent?

Question

A_n = (9n)/(4n+9)

for the series: Sum of (1, infinity) (A_n). is it convergent? (n=1)

Thank you for any help on this problem!

For the sequence {A_n} convergent of divergent?

what would the sum of the series be?

Answer 1

lim A_n = 9/4
n-> infinity

The limit should be 0 so the series diverges.

Answer 2

Geez, I haven't done one of these in a while. Double check my logic.

since n -> Infinity, then the +9 becomes negligible.
So consider 9n/4n, the limit should approach 9/4, thus I would say A_n converges.

Answer 3

The sum is not convergent. As n goes to inf, the series goes to 9/4. The last term must go to zero in order for the sum to converge.

What you end up with is 9/4 being added an infinite number of times. The sum does not converge.

Remember there is a difference between the sum of the series converging, and the last term being finite. It is possible for the last term to be finite and for the sum to not converge.

Answer 4

lim n -> oo (9n)/(4n+9) = lim n -> oo (9)/(4+9/n) = 9/(4 + 0) = 9/4. Since 9/4 <>0, it follows the series Sum of (1, infinity) (A_n) diverges to oo.

Answer 5

If we divide the top and bottom by n, we get the equivalent expression 9 / (4 + (9/n)). The limit of this expression itself goes to 9/4. Since the limit of the expression does not go to zero, then the series Σ9 / (4 + (9/n)) diverges.

This is a case where the limit test proves divergence. However, if the limit was zero, then we'd have to try another test.

Answer 6

Obviously divergent. The terms increase with increasing n, something that can never happen in a convergent series.