# Math Help - Special sequence

1. ## Special sequence

Let $\{a_n\}_{n=1}^{\infty}\subset N$ be strictly monotone increasing sequence

$\displaystyle
\varphi(n)=\,\text{card}\,\{k|\,a_k\leq n\}$
and $\varphi(n)\thicksim\frac n{\ln(n)}\ ,\ n\to\infty$

Prove that

$\sum_{n=1}^\infty \frac 1{a_{a_n}}<+\infty,\ \text{and}\ \,\sum_{n=1}^\infty \frac 1{a_n}=+\infty
$

2. Originally Posted by ns1954
Let $\{a_n\}_{n=1}^{\infty}\subset N$ be strictly monotone increasing sequence

$\displaystyle
\varphi(n)=\,\text{card}\,\{k|\,a_k\leq n\}$
and $\varphi(n)\thicksim\frac n{\ln(n)}\ ,\ n\to\infty$

Prove that

$\sum_{n=1}^\infty \frac 1{a_{a_n}}<+\infty,\ \text{and}\ \,\sum_{n=1}^\infty \frac 1{a_n}=+\infty
$
Are these challenge questions? Are you just going to pose forty-seven million challenge questions without A) specifying whether they are indeed challenge questions B) contributing to MHF?