Tính tổng K=1/5+2/5^2+3/5^3+...+100/5^100

\(linh=\dfrac{1}{5}+\dfrac{2}{5^2}+\dfrac{3}{5^3}+\dfrac{4}{5^4}+....+\dfrac{100}{5^{100}}\)

\(5linh=5\left(\dfrac{1}{5}+\dfrac{2}{5^2}+\dfrac{3}{5^3}+\dfrac{4}{5^4}+...+\dfrac{100}{5^{100}}\right)\)

\(5linh=1+\dfrac{2}{5}+\dfrac{3}{5^2}+\dfrac{4}{5^3}+...+\dfrac{100}{5^{99}}\)

\(5linh-linh=\left(1+\dfrac{2}{5}+\dfrac{3}{5^2}+\dfrac{4}{5^3}+...+\dfrac{100}{5^{99}}\right)-\left(\dfrac{1}{5}+\dfrac{2}{5^2}+\dfrac{3}{5^3}+\dfrac{4}{5^4}+...+\dfrac{100}{5^{100}}\right)\)

\(4linh=1+\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+...+\dfrac{1}{5^{99}}-\dfrac{100}{5^{100}}\)

Đặt:

\(linh_2=1+\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+....+\dfrac{1}{5^{99}}\)

\(5linh_2=5\left(1+\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+....+\dfrac{1}{5^{99}}\right)\)

\(5linh_2=5+1+\dfrac{1}{5}+\dfrac{1}{5^2}+...+\dfrac{1}{5^{98}}\)

\(5linh_2-linh_2=\left(5+1+\dfrac{1}{5}+\dfrac{1}{5^2}+...+\dfrac{1}{5^{98}}\right)-\left(1+\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+...+\dfrac{1}{5^{99}}\right)\)

\(4linh_2=5-\dfrac{1}{5^{99}}\)

\(linh=\dfrac{5}{4}-\dfrac{1}{5^{99}.4}\)

Nên \(4linh=\dfrac{5}{4}-\dfrac{1}{5^{99}.4}-\dfrac{100}{5^{100}}\)

Khi đó \(linh=\dfrac{5}{16}-\dfrac{1}{5^{99}.16}-\dfrac{100}{5^{100}.4}\)

Chứng minh: A=1/5-2/5^2+3/5^3-4/5^4+...+2019/5^2019-2020/5^2020<5/36