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roman
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9 votes
Accepted

Proof that $\lim_{n \to \infty} n[\log (n+1)-\log(n)]=1$

8 votes

An example of a sequence which does not have any subsequence with a finite limit.

6 votes

Values of the limit $\lim\limits_{x\to+\infty}\left(\sqrt{(x+a)(x+b)}-x\right)$

5 votes

Prove that $\lim_{n\to\infty} (\sqrt{n^2+n}-n) = \frac{1}{2}$

4 votes
Accepted

Solving $\lim_{x\to+\infty}{x^2(\cos(\frac{\pi}{x+2})-1)}$

4 votes
Accepted

Evaluate the limit of the sequence: $\lim_{n_\to\infty}\frac{\sqrt{(n-1)!}}{(1+\sqrt{1})\cdot(1+\sqrt{2})\cdot (1+\sqrt{3})\cdots (1+\sqrt{n})}$

3 votes
Accepted

How to prove $\lim_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0 $

3 votes

Determining the minimum value of the function $y = x + 2\sqrt{x^2 - \sqrt{2}x + 1}$

3 votes

Find $\lim_{x \to 0} (\sin x)^{1/x} + (1/x)^{\sin x}$

3 votes

Solving integral $\int \frac{\ln x-1}{\ln^2x}dx$

2 votes

Riemann sum of $1/x^3$

2 votes

Help understanding the steps of a solved limit

2 votes

Monotonicity of $a_n=1+\frac{(-1)^n}{n}$

1 vote

Different methods of evaluating $\int\sqrt{a^2-x^2}dx$:

1 vote

Proof verification that $\{x_n\} = 0,\underbrace{77\dots 7}_{\text{n times}}$ is a Cauchy sequence.

1 vote

Recursive proof that $n^n \geq n!$

1 vote

Prove that $C_{3 \over 2}^n$ is bounded given $C_{a}^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}$

1 vote

How do I prove the identities of these questions?

1 vote

Trouble solving recursive function

1 vote

Compute $ \lim\limits_{n \to \infty}\frac{\sqrt{3n^2+n-1}}{n+\sqrt{n^2-1}}$

1 vote
Accepted

How to solve $\lim \left(\frac{n^3+n+4}{n^3+2n^2}\right)^{n^2}$

1 vote

Why does $\lim\limits_{x\rightarrow0}\frac{1}{2x-1} \log(2^{1+\sin{x}}-1) = 2 $?

1 vote

Error evaluating $ \lim_{x\to 0}\frac{x-\tan x}{x^3} $

1 vote

A sequence $\{a_n\}$ is such that $\lim_n(a_{n+1} - a_n) = 0$. Given some additional properties of $a_n$ prove that it converges.

1 vote

Riemann sum of $\int_1^2 {1\over x^2} dx$.

0 votes

Prove that $\lim_{n \to\infty}(1+\frac{1}{n})^n =\lim_{n \to\infty} (\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots + \frac{1}{n!})$

0 votes

Show that $\lim_{n}\sum_{k=n}^{2n}{1\over k} = \ln2$ using elementary methods.

0 votes

Please help with $\sum\limits_{r=1}^{N}\frac{2r+3}{3^r(r+1)}$

0 votes

How do you show $\frac{\epsilon \sin(\epsilon t)}{1-\epsilon t } = \mathcal{O(\epsilon ^2)} \quad (1 \leq t \leq 1000)$

0 votes
Accepted

Proving that a sequence converges if $|a_n - a_{n+1}| < Mr^n$ for some $M > 0$ and $r \in (0,1).$