Prove that the sequence $b_n=\left(1+\frac{1}{n}\right)^{n+1}$ is decreasing
92175
Given $y_n=(1+\frac{1}{n})^{n+1}$ show that $\lbrace y_n \rbrace$ is a decreasing sequence
67650
Short and intuitive proof that $\left(\frac{n}{k}\right)^k \leq \binom{n}{k}$
80826
To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$
46830
Prove that the sequence $a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$ is convergent and find its limit
136274
$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
70561
limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$
61212
Using BMCT to prove the sequence $a_{n+1} = \sqrt{2 + a_n}$ is bounded and increasing.
71547
Prove $ 1+\left( 1-\frac{x}{n}\right)+...+\left(1-\frac{x}{n}\right)^{n-1}=\frac{n}{x}\left(1-\left(1-\frac{x}{n}\right)^n\right)$
102523
Show that $\lim\limits_{n \to \infty} \sup\limits_{k \geq n} \left(\frac{1+a_{k+1}}{a_k}\right)^k \ge e$ for any positive sequence $\{a_n\}$
91914
Prove that for the Fibonacci sequence $(F_n)$, $F_n$ divides $F_{2n}$.
116414
Proving that the difference of the terms of a sequence, multiplied by $n$ converge
38002
how to prove that this positive sequence converges to 1
145863
sequence of positive numbers satisfying $a_{n+1}=\frac{2}{a_n+a_{n-1}}$, prove it converges
49497
$\lim a_n = 0$ if $\left|\frac{a_{n +1}}{a_n}\right|\to L < 1$
5072
To evaluate limit of sequence $\left(\left( 1 + \frac1n \right) \left( 1 + \frac2n \right)\cdots\left( 1 + \frac nn \right) \right)^{1/n}$
150000
Convergence of the sequence $\frac{1}{n\sin(n)}$
123523
Showing $\left(1+\frac{1}{n}\right)^n$ is an increasing sequence by comparing the binomial expansions term by term