# Thread: Ratio test for series divergence help.

1. ## Ratio test for series divergence help.

This is from so called ratio test, which says

The series Ʃa(n) diverges if abs{a(n+1)/a(n)} ≥ 1 for n≥N where N is some fixed integer.

I was wondering if I could replace the condition with lim inf abs{a(n+1)/a(n)} ≥ 1.

So basically what I'm asking is whether these two below are equivalent or not.
1. abs{a(n+1)/a(n)} ≥ 1 for n≥N where N is some fixed integer
2. lim inf abs{a(n+1)/a(n)} ≥ 1

*a(n) means nth term of the sequence, and abs means absolute value.

2. ## Re: Ratio test for series divergence help.

I think You couldn't. Supposing that is...

$\displaystyle \lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}}|=1$ (1)

... the behaviour of $\displaystyle |\frac{a_{n+1}}{a_{n}}}|$ can be 'oscillating' around 1 and in this case it isn't $\displaystyle |\frac{a_{n+1}}{a_{n}}}| \ge 1$ 'for $\displaystyle n \ge N$ where N is some fixed integer'...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$

3. ## Re: Ratio test for series divergence help.

Originally Posted by chsoviz0716
This is from so called ratio test, which says

The series Ʃa(n) diverges if abs{a(n+1)/a(n)} ≥ 1 for n≥N where N is some fixed integer.

I was wondering if I could replace the condition with lim inf abs{a(n+1)/a(n)} ≥ 1.

So basically what I'm asking is whether these two below are equivalent or not.
1. abs{a(n+1)/a(n)} ≥ 1 for n≥N where N is some fixed integer
2. lim inf abs{a(n+1)/a(n)} ≥ 1

*a(n) means nth term of the sequence, and abs means absolute value.

Actually, the series is only convergent where \displaystyle \displaystyle \begin{align*} \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| < 1 \end{align*} and divergent where \displaystyle \displaystyle \begin{align*} \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| > 1\end{align*}.

The ratio test is INCONCLUSIVE if \displaystyle \displaystyle \begin{align*} \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| = 1 \end{align*}.

4. ## Re: Ratio test for series divergence help.

The series diverges if $\displaystyle \lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}|=1$ but is also $\displaystyle \forall{n>N}\ |\frac{a_{n+1}}{a_{n}}|\ge 1$...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$

5. ## Re: Ratio test for series divergence help.

Originally Posted by chisigma
I think You couldn't. Supposing that is...

$\displaystyle \lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}}|=1$ (1)

... the behaviour of $\displaystyle |\frac{a_{n+1}}{a_{n}}}|$ can be 'oscillating' around 1 and in this case it isn't $\displaystyle |\frac{a_{n+1}}{a_{n}}}| \ge 1$ 'for $\displaystyle n \ge N$ where N is some fixed integer'...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$
Thank you for the reply first.
The example you took shows even if 2 is satisfied, 1 need not be satisfied, so clearly they are not equivalent.
What about the other way around?
Can I conclude that if 1 is satisfied, 2 is satisfied?
or is there an example which satisfies 1 but doesn't satisfy 2?

6. ## Re: Ratio test for series divergence help.

Can I conclude that if 1 is satisfied, 2 is satisfied?... or is there an example which satisfies 1 but doesn't satisfy 2? ...

Let's indicate $\displaystyle \rho_{n}= \frac{a_{n+1}}{a_{n}}$ and consider the possibility $\displaystyle \rho_{n}=2+(-1)^{n}$. Clearly 1) is satisfied and 2) is not because the limit for n tending to infinity of $\displaystyle \rho_{n}$ doesn't exist...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$

7. ## Re: Ratio test for series divergence help.

Originally Posted by chisigma
Can I conclude that if 1 is satisfied, 2 is satisfied?... or is there an example which satisfies 1 but doesn't satisfy 2? ...

Let's indicate $\displaystyle \rho_{n}= \frac{a_{n+1}}{a_{n}}$ and consider the possibility $\displaystyle \rho_{n}=2+(-1)^{n}$. Clearly 1) is satisfied and 2) is not because the limit for n tending to infinity of $\displaystyle \rho_{n}$ doesn't exist...
Oh may be you misunderstood what I wrote as 'lim inf'. I didn't mean limit as n goes to infinity but meant 'limit inferior' which is defined to be the infimum of all subsequential limits of the original sequence. I attach the link here.
Limit superior and limit inferior - Wikipedia, the free encyclopedia

So even in the case of 2+(-1)^n, lim inf abs{a(n+1)/a(n)} ≥ 1 still holds. In fact, it equals one in this case.

8. ## Re: Ratio test for series divergence help.

For the sequence $\displaystyle \rho_{n}=\frac{|a_{n+1}|}{|a_{n}|}= 2+(-1)^{n}$ it doesn't exist $\displaystyle \lim_{n \rightarrow \infty} \rho_{n}$...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \chi$