nice inequality

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July 28th 2010, 08:43 AM
fxs12

nice inequality

$ \begin{array}{l} prove\;that \\ \\ 1 \prec \frac{1}{{1001}} + \frac{1}{{1002}} + \frac{1}{{1003}} + ...... + \frac{1}{{3009}} \prec 1\frac{1}{3} \\ \end{array} $
July 28th 2010, 10:15 AM
Archie Meade

Quote:

Originally Posted by fxs12

$ \begin{array}{l} prove\;that \\ \\ 1 \prec \frac{1}{{1001}} + \frac{1}{{1002}} + \frac{1}{{1003}} + ...... + \frac{1}{{3009}} \prec 1\frac{1}{3} \\ \end{array} $

Euler's constant is

$\displaystyle\lim_{n\rightarrow\infty}\left(\frac{ 1}{1}+\frac{1}{2}+\frac{1}{3}+......+\frac{1}{n}-log_en\right)$

$log_en\approx\left(1+\frac{1}{2}+\frac{1}{3}+....+ \frac{1}{n}\right)-Euler's\ constant$

$\frac{1}{1001}+\frac{1}{1002}+.......+\frac{1}{300 9}\approx\ log_e3009-log_e1000\approx\ log_e3.009$

$e=2.718...$

$e^{\frac{4}{3}}=3.79...$

Therefore

$log_e2.718..<log_e3.009<log_e3.79..$
July 29th 2010, 02:27 AM
fxs12

thank you , but how didi you get this?

$\log _e n \approx (\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + ....... + \frac{1}{n}) - Euler's\;cons\tan t$
July 29th 2010, 01:16 PM
Archie Meade

Sorry, that approximation may be a bit on the crude side!!

What you have is the difference between a pair of Harmonic series.

There are alternative quick calculations for that,
and you may be requiring a particular type of solution,
possibly along other lines.

$H_n=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+.......... ......+\frac{1}{n}$

One approximation is...

$\frac{1}{2(n+1)}+log_en+\gamma<H_n<\frac{1}{2n}+lo g_en+\gamma$

where $\gamma\approx\ 0.5772156649$ is Euler's constant.

Taking either $\frac{1}{2n}$ or $\frac{1}{2n+1}$

we get approximately $H_{3009}-H_{1000}\approx\ \frac{1}{2(3009)}+log_e3009-\left(\frac{1}{2(1000)}+log_e1000\right)$

or

$\frac{1}{2(3010)}+log_e3009-\left(\frac{1}{2(1001)}+log_e1000\right)$

which is basically approximately the difference between the logarithms