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relations between the root test and the ratio test

I know the theorem is correct if they are exist $$ \lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} \leq \lim\inf\limits_{n\rightarrow \infty} (A_n)^{1/n} \leq \lim\sup\limits_{n\rightarrow \infty} (A_n)^{1/n} \leq \lim\sup\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} $$

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Here is the 1st question.

If $$\lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} $$ and $$ \lim\sup\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} $$ are $\infty$

then, $$ \lim\inf\limits_{n\rightarrow \infty} (A_n)^{1/n} $$ and $$ \lim\sup\limits_{n\rightarrow \infty} (A_n)^{1/n} $$ are $\infty$?

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And 2nd question is $$ \lim_{n\rightarrow \infty} \frac{|A_{n+1}|}{|A_n|} = \infty $$ then $\lim_{n\rightarrow \infty} (A_n)^{1/n} = \infty$

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Actually, 2nd question looks like easy, but I can't prove yet.

Could you please help me?

Thanks

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1 Answer 1

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Both tests are for series with positive terms; or you should put absolute values around $A_n$ and $A_{n+1}$ everywhere. Otherwise you'll have a problem taking the roots. And division by $0$ is generally frowned upon.

Assuming all the terms are positive, it is true that $$\liminf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} \leq \liminf\limits_{n\rightarrow \infty} (A_n)^{1/n} \leq \limsup\limits_{n\rightarrow \infty} (A_n)^{1/n} \leq \limsup_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}$$ in the extended sense: the limits are allowed to take value $+\infty$. (By the way, this is common with $\liminf $ and $\limsup$ anyway.

The proof is not really different from the finite case. Here's a proof of the first inequality. Let $b$ be a number such that $b< \liminf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} $. Then we have $A_{n+1}>bA_n$ for all sufficiently large $n$. Applying this iteratively, you'll get a lower bound of the form $A_n\ge C b^n$ for all sufficiently large $n$. This implies $\liminf\limits_{n\rightarrow \infty} (A_n)^{1/n} \ge b$. Since $b$ was an arbitrary number such that $b< \liminf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} $, the conclusion $\liminf\limits_{n\rightarrow \infty} (A_n)^{1/n}\ge \liminf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}$ follows.

Note that it makes no difference whether these limits are finite or not; the argument works the same.

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  • $\begingroup$ Thanks for your answer and sorry for lating response. But I have some qeustion. // In my opinion, you used argument that "If b < A implies b < B for any b, then A=<B", right? // How can I justify this argument? Thanks sincerely:) $\endgroup$
    – user143993
    Jun 8, 2014 at 6:12
  • $\begingroup$ @user143993 Suppose the contrary: $A>B$. Take $b$ between $A$ and $B$. Then $b<A$, but $b>B$. $\endgroup$
    – user147263
    Jun 8, 2014 at 6:17
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    $\begingroup$ I understand well. THANKS for your teaching. :-) $\endgroup$
    – user143993
    Jun 8, 2014 at 6:31

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