Does the Ratio Test Confirm Divergence for This Series?

[ani9890]

Ratio Test, SUPER URGENT, help?

Consider the series
∑ n=1 to infinity of asubn, where asubn = [8^(n+4)] / [(8n^2 +7)(5^n)]
use the ratio test to decide whether the series converges. state what the limit is.

From the ratio test I got the limit n-> infinity of
[8^(n+1+4)] / [(8(n+1)^2 +7)(5^n+1)] / [8^(n+4)] / [(8n^2 +7)(5^n)]
= [8^n+5(8n^2 + 7)5^n] / (8(n+1)^2 +7)5^(n+1) (8^n+4)
=[8^n(8^5)(8n^2 + 7) 5^n] / (8n^2 + 16n + 15)(5^n)(5)(8^n)(8^4)

the lim n-> infinity of [262144n^2 +229376] / [163840n^2 + 327680n + 307200]
= 8/5
which is divergence. Is this correct?

Help?

 

[SammyS]

Consider the series
∑ n=1 to infinity of asubn, where asubn = [8^(n+4)] / [(8n^2 +7)(5^n)]
use the ratio test to decide whether the series converges. state what the limit is.

Is this the series you're working with ?

$\displaystyle \sum_{n=1}^{\infty} \frac{8^{n+4}}{(8n^2 +7)(5^n)}$

From the ratio test I got the limit n-> infinity of
[8^(n+1+4)] / [(8(n+1)^2 +7)(5^n+1)] / [8^(n+4)] / [(8n^2 +7)(5^n)]
= [8^n+5(8n^2 + 7)5^n] / (8(n+1)^2 +7)5^(n+1) (8^n+4)
=[8^n(8^5)(8n^2 + 7) 5^n] / (8n^2 + 16n + 8)(5^n)(5)(8^n)(8^4)

the lim n-> infinity of (32775) / 16n + 4109 = 0
which is absolute convergence. But that answer is wrong.

Help?

Are you saying your result is wrong because the series doesn't converge absolutely?

 

[ani9890]

yes, that is the series I'm working with.

And actually, I've redone the problem and edited my work above.
Now I got a limit of 8/5 = 1.6 so the series diverges.
Is this correct (did I calculate the limit correct)?

 

[Dick]

yes, that is the series I'm working with.

And actually, I've redone the problem and edited my work above.
Now I got a limit of 8/5 = 1.6 so the series diverges.
Is this correct (did I calculate the limit correct)?

Yes, that would be correct.