• Feb 1st 2010, 12:26 PM
ha11
Question in the Attach Files, thanks(Wink)
• Feb 1st 2010, 04:58 PM
hatsoff
Quote:

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

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

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

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

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

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

$=\sum_{i=1}^{\infty}\frac{1}{i}-1$.
• Feb 1st 2010, 11:59 PM
ha11
thanks my friend.