# What if a ratio test of series = 1

#### orendacl

SUM, from 1 to infinity of:

ln(n)
________
(n+1)^3

I work this out using the ratio test for convergence and the answer I get is "1"...

Does this sound right?
If so, this then means that there are no other tests that can be done correct? The test is simply inconclusive.
Or:
Is there another test that should be performed next?

#### Prove It

MHF Helper
SUM, from 1 to infinity of:

ln(n)
________
(n+1)^3

I work this out using the ratio test for convergence and the answer I get is "1"...

Does this sound right?
If so, this then means that there are no other tests that can be done correct? The test is simply inconclusive.
Or:
Is there another test that should be performed next?
You could try the root test...

Root test - Wikipedia, the free encyclopedia

#### Failure

SUM, from 1 to infinity of:

ln(n)
________
(n+1)^3

I work this out using the ratio test for convergence and the answer I get is "1"...

Does this sound right?
No, it doesn't. The ratio test acually tells you that this series converges absolutely:

$$\displaystyle \frac{a_{n+1}}{a_n}=\frac{\ln(n+1)}{\ln (n)}\cdot\frac{n^3}{n^3+3n^2+3n+1}\rightarrow 0$$

The first factor goes to 1 as n goes to $$\displaystyle \infty$$, since

$$\displaystyle \frac{\ln(n+1)}{\ln (n)}=\frac{\ln(n)+\ln(1+1/n)}{\ln(n)}$$

and the second factor goes to 0.

If so, this then means that there are no other tests that can be done correct?
No: if the ratio test fails, the root test might still be used to prove absolute convergence.

#### simplependulum

MHF Hall of Honor
No, it doesn't. The ratio test acually tells you that this series converges absolutely:

$$\displaystyle \frac{a_{n+1}}{a_n}=\frac{\ln(n+1)}{\ln (n)}\cdot\frac{n^3}{n^3+3n^2+3n+1}\rightarrow 0$$

The first factor goes to 1 as n goes to $$\displaystyle \infty$$, since

$$\displaystyle \frac{\ln(n+1)}{\ln (n)}=\frac{\ln(n)+\ln(1+1/n)}{\ln(n)}$$

and the second factor goes to 0.

No: if the ratio test fails, the root test might still be used to prove absolute convergence.

Doesn't it $$\displaystyle \frac{\ln(n+1)}{\ln (n)}\cdot\frac{n^3}{n^3+3n^2+3n+1}$$ go to $$\displaystyle 1$$ ?

We can use this inequality

$$\displaystyle \ln{n} < n < n+1$$

If a ratio test of series = 1 , it means nothing .

#### Failure

Doesn't it $$\displaystyle \frac{\ln(n+1)}{\ln (n)}\cdot\frac{n^3}{n^3+3n^2+3n+1}$$ go to $$\displaystyle 1$$ ?
Oops, yes, you are right (obviously I didn't even take the time to read carefully enough what I had written myself).

Maybe it was because I was already quite sure that the series converges. (Why? - Because I know that $$\displaystyle \sum_n\frac{1}{n^2}$$ converges and can be used to bound the given series from above.)

If a ratio test of series = 1 , it means nothing .
Sure, in that case, the ratio test does not decide the question of convergence vs. divergence.

#### Jester

MHF Helper
Try using the fact that for $$\displaystyle n > 0$$ that

$$\displaystyle \frac{\ln n}{(n+1)^3} < \frac{\ln n}{n^3} < \frac{n}{n^3} = \frac{1}{n^2}$$