Tinh 2S, roi lay 2S-S=1-1/2^100

ban co the giai thich cu the hon duoc khong?

=)2S-S=(1+1/2+1/2^2+......+1/2^99)-(1/2+1/2^2+1/2^3+......+1/2^100)

=)S=1-1/2^100

=)S=2^100-1/2^100

Cứ chép vào đáp ản chuẩn nhớ là ko cần tính rõ ra=)))

\(S=1+\frac{1}{3}+\frac{1}{3^2}+........+\frac{1}{3^n}\)

\(3S=3+1+\frac{1}{3}+.......+\frac{1}{3^{n-1}}\)

\(\Rightarrow3S-S=\left(3+1+\frac{1}{3}+......+\frac{1}{3^{n-1}}\right)-\left(1+\frac{1}{3}+\frac{1}{3^2}+......+\frac{1}{3^n}\right)\)

\(\Rightarrow2S=3-\frac{1}{3^n}\Rightarrow2S=\frac{3^{n+1}-1}{3^n}\Rightarrow S=\frac{3^{n+1}-1}{2.3^n}\)

đk: x>=0; x khác 3

a) \(P=\frac{\sqrt{x}-3}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-3\right)}-\frac{5}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-3\right)}+\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{x}-3}=\frac{\sqrt{x}-3-5+x-4}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-3\right)}=\frac{x+\sqrt{x}-12}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-3\right)}\)

\(P=\frac{\left(\sqrt{x}+4\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-3\right)}=\frac{\sqrt{x}+4}{\sqrt{x}+2}\)

b) \(P=\frac{\sqrt{x}+2+2}{\sqrt{x}+2}=1+\frac{2}{\sqrt{x}+2}\)

ta có: \(x\ge0\Rightarrow\sqrt{x}\ge0\Leftrightarrow\sqrt{x}+2\ge2\Leftrightarrow\frac{2}{\sqrt{x}+2}\le1\Leftrightarrow1+\frac{2}{\sqrt{x}+2}\le2\Rightarrow MaxP=2\Rightarrow x=0\)

Đặt \(A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\)

\(\Rightarrow2A=2+1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2011}}\)

            \(A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2011}}+\frac{1}{2^{2012}}\)

\(\Rightarrow2A-A=A=2-\frac{1}{2^{2012}}\)

Áp dụng a/(a^4+a^2+1)=1/2.(1/(a^2-a+1)-1/(a^2+a+1)) ta được

A=1/2.(1/(1^2-1+1)-1/(1^2+1+1)+1/(2^2-2+1)-1/(2^2+2+10)+...+1/(2014^2-2014+1)-1/(2014^2+2014+1))

A=1/2.(1-1/(2014^2+2014+1))

A=-2029105/4058211

(CHẮC CHẮN ĐÚNG)

A=2029105/4058211

giup minh voi cac ban