Linked Questions

5
votes
3answers
8k views

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://...
4
votes
2answers
5k views

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 ...
7
votes
3answers
4k views

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 ...
5
votes
3answers
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 ...
3
votes
1answer
2k views

Show root test is stronger than ratio test [duplicate]

Let $a_n$ be a sequence of real positive numbers. Show $$\liminf_{n\to  \infty} \frac{a_{n+1}}{a_n} \leq \liminf_{n\to \infty}\,(a_n)^{1/n}  \leq \limsup_{n\to \infty} \, (a_n)^{1/n} \leq \limsup_{n\...
3
votes
1answer
1k views

Finding limit using inequalities: $\liminf \frac{a_{n+1}}{a_n} \le \liminf (a_n)^ {1/n}\le\limsup (a_n)^ {1/n}\le \limsup \frac{a_{n+1}}{a_n}$ [duplicate]

The purpose of this exercise is to prove that $\lim \frac{n}{(n!)^{1/n}}=e$ when $n$ goes to infinity. In order to find the limit, the following inequality is used when $n$ goes to infinity with ${...
2
votes
1answer
395 views

Proof of limit inequality [duplicate]

Prove that for any sequence $\{x_n\}$ of positive real numbers $$\lim\text{sup}\sqrt[n]{x_n}\leq \lim\text{sup}\frac{x_{n+1}}{x_n}.$$ My attempt: Let $A = \lim\text{sup}\frac{x_{n+1}}{x_n}$. ...
1
vote
1answer
377 views

Show that $\limsup|s_n|^{1\over n}\le \limsup|{s_{n+1}\over s_n}|$ [duplicate]

Possible Duplicate: Inequality involving $\limsup$ and $\liminf$ limit of $\frac{a_{n+1}}{a_n}$ Show that $\limsup|s_n|^{1\over n}\le \limsup|{s_{n+1}\over s_n}|$ and similarly $\liminf|s_n|^{1\...
2
votes
1answer
210 views

The inequality $\limsup \frac{c_{n+1}}{c_n} \geqslant \limsup \sqrt[n]{c_n}$ [duplicate]

$c_n>0$ in $\mathbb{R}$, prove$$\limsup \frac{c_{n+1}}{c_n} \geqslant \limsup \sqrt[n]{c_n}$$ Tried to present $\sqrt[n]{c_n}=c_1 \frac{c_2}{c_1}...\frac{c_{n+1}}{c_n}$. Seemingly $\sqrt[n]{c_n}$ ...
4
votes
2answers
232 views

Inequality between limits inferior [duplicate]

Let $(a_n)$ be a non-zero real sequence with $\left (\frac{a_{n+1}}{a_n} \right)$ bounded. How might we prove that $$\liminf_{n\to\infty} \, \frac{|a_{n+1}|}{|a_n|} \leq \liminf_{n\to\infty} \, |a_n|^{...
0
votes
1answer
125 views

$\lim \frac{|a_{n+1}|}{|a_n|} = \lim \sqrt[n]{|a_n|}=L$. [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 $\lim \sqrt[n]{|a_n|}=L$.
0
votes
2answers
62 views

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 $\alpha=\limsup_{n\rightarrow\infty}\sqrt[n]{|c_{n}|}$ ...
2
votes
2answers
98 views

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 ...
1
vote
1answer
53 views

An inequality involving limit inferior and limit superior [duplicate]

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 \...
3
votes
0answers
32 views

Comparison of limsup of two particular sequences [duplicate]

I'm reviewing my analysis, and have found the following problem on an old exam. I don't know how to solve it, so I'm hoping that, if someone can explain it to me, I can gain some familiarity with a ...

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