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### Root test is stronger than ratio test? [duplicate]

I am a little bit confused regarding the meaning of the phrase :" Root test is stronger than ratio test", and was hoping you will be able to help me figure it out. As far as I can see here: https://...
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### Given that $a_n > 0$, prove: $\liminf \left(\frac{a_{n+1}}{a_n}\right)\leq \liminf\; \sqrt[n]{a_n}\leq\limsup\left(\frac{a_{n+1}}{a_n}\right)$ [duplicate]

Given that $a_n > 0$, I need to prove: $\liminf \left(\dfrac{a_{n+1}}{a_n}\right)\;\leq \;\liminf\; \sqrt[\Large n]{a_n}\;\leq\;\limsup\left(\dfrac{a_{n+1}}{a_n}\right)$. I am really confused ...
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### Do the sequences from the ratio and root tests converge to the same limit? [duplicate]

For example, if we have: $$\sum_{n=1}^{\infty}a_n$$ Where the ratio test is satisfied. That is $\exists L$ s.t. $$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L < 1$$ Does this ...
4k views

### Ratio test and the Root test [duplicate]

Both the ratio test and the root test define a number (via a limit). If both limits exist (and shows that the series is convergent), what (if any) is the relation between the 2 numbers ? are they ...
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### $\lim \frac{|a_{n+1}|}{|a_n|} = \lim \sqrt[n]{|a_n|}=L$. [duplicate]

How can I prove that if $\sum a_n$ is a series with elements non-null and $\lim \frac{|a_{n+1}|}{|a_n|} = L$, so $\lim \sqrt[n]{|a_n|}=L$.
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### Is $\limsup_{n\rightarrow\infty}\frac{c_{n+1}}{c_{n}}=\limsup_{n\rightarrow\infty}\sqrt[n]{c_{n}}$ for a positive sequence $(c_{n})_{n\geq 1}$? [duplicate]

I was going through Theorem 3.39 from Baby Rudin page no 69. I will restate the theorem, Given the power series $\sum_{n\geq0}c_{n}z^{n}$, put $\alpha=\limsup_{n\rightarrow\infty}\sqrt[n]{|c_{n}|}$ ...
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### Convergence issues in the identity $\lim a_n^{1/n}=\lim a_{n+1}/a_n$. [duplicate]

I discovered that, for the convergence/divergence of series, the ratio test is stronger than the n-th root test. See here for more details. Since both tests should give rise to the same radius of ...
I have the following assertion to be proved for $a_n > 0, \forall n \in \mathbb{N}$:  \underset{n \rightarrow \infty}{\mathrm{lim \inf}} \ \frac{a_{n+1}}{a_n} \leq \underset{n \rightarrow \...