Question

Se considera nr. s= 1+$\frac{1}{ 2^{1} } + \frac{1}{ 2^{2} } + \frac{1}{ 2^{3} } + .... + \frac{1}{ 2^{2009} }$. Demonstrati ca s ∈ (1,2)

Answer 1

s=1+1/2+1/2^2+1/2^3+...+1/2^2009
s/2=1/2+1+1/2^2+1/2^3+...+1/2^2009+1/2^2010-1=s+1/2^2010-1
s/2=1-1/2^2010
s=2-1/2^2009

2>s=2-1/2^2009>1, deci s apartine (1,2).