Chứng minh rằng : 1/2^2+1/2^3+1/2^4+1/2^5+…+1/2^10 <1 25/10/2021 Bởi Valentina Chứng minh rằng : 1/2^2+1/2^3+1/2^4+1/2^5+…+1/2^10 <1
Đáp án: Đặt A=$\frac{1}{2^2} +\frac{1}{2^3}+…….+\frac{1}{2^{10}}$ $⇒2A=\frac{1}{2} +\frac{1}{2^2}+…….+\frac{1}{2^{9}}$ Lấy $2A-A$ =$(\frac{1}{2} +\frac{1}{2^2}+…….+\frac{1}{2^{9}})-(\frac{1}{2^2} +\frac{1}{2^3}+…….+\frac{1}{2^{10}})$ $⇒A=\frac{1}{2}-\frac{1}{2^{10}}<\frac{1}{2}<1$ $⇒A<1$ Bình luận
Đáp án:
Đặt A=$\frac{1}{2^2} +\frac{1}{2^3}+…….+\frac{1}{2^{10}}$
$⇒2A=\frac{1}{2} +\frac{1}{2^2}+…….+\frac{1}{2^{9}}$
Lấy $2A-A$ =$(\frac{1}{2} +\frac{1}{2^2}+…….+\frac{1}{2^{9}})-(\frac{1}{2^2} +\frac{1}{2^3}+…….+\frac{1}{2^{10}})$
$⇒A=\frac{1}{2}-\frac{1}{2^{10}}<\frac{1}{2}<1$
$⇒A<1$