# Special sequence

• January 21st 2010, 08:29 AM
ns1954
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
$
• January 21st 2010, 01:50 PM
Drexel28
Quote:

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
$

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