Please help me

**ha11** · February 1st 2010, 11:26 AM

Question in the Attach Files, thanks

**hatsoff** · February 1st 2010, 03:58 PM

Originally Posted by ha11

Show that $\sum_{k=1}^n\frac{1}{k}=O(\log n)$

and $\int_1^{\infty}\frac{\lfloor t\rfloor}{t^2}dt=\sum_{1\leq r\leq t\leq n}\int_1^{\infty}\frac{dt}{t^2}=\sum_{r=1}^n\left( \frac{1}{r}-\frac{1}{n}\right)$ .

It may help others to see this without the attachment.

Now, as for solutions.... regarding the second, where does $n$ come from? I can only show that

$\int_1^{\infty}\frac{\lfloor t\rfloor}{t^2}dt=\sum_{i=1}^{\infty}\int_i^{i+1}\f rac{i}{t^2}dt$

$=\sum_{i=1}^{\infty}i\int_i^{i+1}t^{-2}dt$

$=\sum_{i=1}^{\infty}i\left[-t^{-1}\right]_i^{i+1}$

$=\sum_{i=1}^{\infty}i\left(i^{-1}-(i+1)^{-1}\right)$

$=\sum_{i=1}^{\infty}\frac{1}{i+1}$

$=\sum_{i=1}^{\infty}\frac{1}{i}-1$ .

**ha11** · February 1st 2010, 10:59 PM

thanks my friend.

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