# Special sequence

• Jan 21st 2010, 08:29 AM
ns1954
Special sequence
Let $\displaystyle \{a_n\}_{n=1}^{\infty}\subset N$ be strictly monotone increasing sequence

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

Prove that

$\displaystyle \sum_{n=1}^\infty \frac 1{a_{a_n}}<+\infty,\ \text{and}\ \,\sum_{n=1}^\infty \frac 1{a_n}=+\infty$
• Jan 21st 2010, 01:50 PM
Drexel28
Quote:

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

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

Prove that

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

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?