In a sequence of positive terms, $\lim \frac{a_{n+1}}{a_n} = L \Rightarrow \lim (a_n)^{\frac{1}{n}} = L$.
42329
Convergence of Ratio Test implies Convergence of the Root Test
22470
Prove $\lim_{n\to\infty} \frac{a_n}{n}$ exists for positive sequence where $a_{n+m} \leq a_n + a_m$
25294
Monotonicity of the sequences $\left(1+\frac1n\right)^n$, $\left(1-\frac1n\right)^n$ and $\left(1+\frac1n\right)^{n+1}$
68609
Let $a_{n+1}=\sqrt{a_1+a_2+\cdots+a_n}$ .Prove that $ \lim\limits_{n \rightarrow \infty} \frac{a_n}{n}=\frac{1}{2}$
94910
How does one show that $\lim_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x$ converges?
14129
How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$
7507
Proof that $\lim{a_n}=L$ when $n$ goes to infinity, then $\{a_n\}$ its a Cauchy Sequence
25206
How to prove that $\lim\limits_{n\to \infty}a_{n+1}=\lim\limits_{n\to \infty}a_{n}$
73812
Does $\lim \limits_ {n\to \infty}\ \frac{n}{n+1}$ equal $1$?
35885
Monotonicity and convergence of the sequence $a_n=\sum_{k=1}^{n}\frac{1}{k+n}$
135649
How is $\lim_{x \to a}\left(\frac{x^n - a^n}{x - a}\right) = n\times a^{n-1}$?
54513
Prove $\frac 1{x+1}<\ln\left(\frac{x+1}x\right)<0.5\left(\frac 1x+\frac 1{x+1}\right)$
49497
$\lim a_n = 0$ if $\left|\frac{a_{n +1}}{a_n}\right|\to L < 1$
80350
$\limsup \frac{a_n}{b_n}0$ s.t $a_n\le Mb_n$
95991
How to solve $\lim_{x \rightarrow 0} {\frac{(1+x)^a-1}{x}}$?
11879
If $\lim_{n\rightarrow\infty} f(n+1) - f(n) = L$, prove that $\lim_{n\rightarrow\infty} f(n)/n = L$
152132
Proving if $\lim_{n\rightarrow\infty}a_n=L $ then $\lim_{n\rightarrow\infty} \frac{a_1+a_2+\cdots+a_n}n=L $