Prove that $\sum_{x\le{n}}\frac{1}{\varphi{(x)}} = \mathcal{O}(ln{}(n))$
47790
If $x \geq 2$ prove that $\sum_{ n \leq x} \frac{1}{\phi(n)} = \mathcal{O}(\log(x))$
8913
Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$
32510
Proving the convergence of $\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$
349
Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$
3110
Prove $\frac{1}{\sqrt{x}}\geq \frac{\ln x}{x-1}$
137640
Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$
134152
Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$
20317
Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$
150565
Prove $(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}$
15164
Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
96961
A proof for the identity $\sum_{n=r}^{\infty} {n \choose r}^{-1} = \frac{r}{r-1}$
16253
Proof that $|\sin(nx)| \le n|\sin(x)|$
135700
Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists
122115
Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
90749
Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.
135219
On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x) $
150563
Find the sum $\sum_{n=1}^{\infty} \frac{\ln n}{n(n- \ln n)}$