Question

Tinh: -1 - 1/2 - 1/4 - 1/8 - ... - 1/1024 .

Answer 1

\(A=-1-\dfrac{1}{2}-\dfrac{1}{4}-\dfrac{1}{8}-....-\dfrac{1}{1024}\)

\(A=-1-\left(\dfrac{1}{2^1}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{10}}\right)\)
Đặt:

\(B=\dfrac{1}{2^1}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{10}}\)

\(2B=2\left(\dfrac{1}{2^1}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{10}}\right)\)

\(2B=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^9}\)

\(2B-B=\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^9}\right)-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{10}}\right)\)

\(B=1-\dfrac{1}{2^{10}}\)

Thay vào \(A\)

\(A=-1-\left(1-\dfrac{1}{2^{10}}\right)\)

\(A=-1-1+\dfrac{1}{2^{10}}\)

\(A=-2+\dfrac{1}{2^{10}}\)