# Deduce the inequality

• September 16th 2009, 02:12 AM
bobred
Deduce the inequality
Hi

If this is the wrong section of the forum to post this I apologise, if so could you tell me which one would be more appropriate?

The answer is probably staring me in the face, but here goes.

I have shown that

$\sum_{r=1}^{n}\frac{r}{2^r}=2-\frac{n+2}{2^n}$ for $n=1,2,3,...$

Is true by induction, but I am then asked to deduce that

$\sum_{r=1}^{n}\frac{r}{2^r}<2$ for $n=1,2,3,...$

Can someone please point me in the right direction.

Thanks
• September 16th 2009, 02:45 AM
Plato
Quote:

Originally Posted by bobred
I have shown that
$\sum_{r=1}^{n}\frac{r}{2^r}=2-\frac{n+2}{2^n}$ for $n=1,2,3,...$
Is true by induction, but I am then asked to deduce that

$\sum_{r=1}^{n}\frac{r}{2^r}<2$ for $n=1,2,3,...$

$2-\frac{n+2}{2^n} <2$ because $-\frac{n+2}{2^n} <0$