# analysis problem

• October 11th 2009, 11:15 AM
IIT 2010
analysis problem
For a positive integer
n , let
an=1+(1/2)+(1/3)+..........+(1/(2^n -1))
then
a) a200<=100
b) a200 >100

• October 12th 2009, 08:47 AM
Enrique2
Here it is underlying the original proof of the divergence of the harmonic series.

Define
$s_1=1>1/2$
$s_2=1/2$
$s_3=1/3+1/4>2(1/4)=1/2$
$s_4 =1/5+1/6+1/7+1/8>4(1/8)=1/2$
$s_5=1/9+\cdots+1/16>8(1/16)=1/2$

$
s_n=(1/(2^{n-2}+1)+\cdots+1/2^{n-1}>(2^{n-1}-2^{n-2})1/2^{n-1}=1/2
$

$
a_n=s_1+\cdots+s_{n}>n(1/2)
$

Now it follows that the harmonic series diverges, and also that your correct answer is b)!