giúp mình nhé!!
\(S=\dfrac{1}{7^2}+\dfrac{2}{7^3}+\dfrac{3}{7^3}+...+\dfrac{69}{7^{70}}\)
\(S=\dfrac{1+2+3+...+69}{\left(7\right)^{2+3+4+...+70}}\)
\(S=\dfrac{\left(69-1\right)+1}{\left(7\right)^{\left(70-2\right)+1}}\)
\(S=\dfrac{69}{7^{69}}\)
\(\Rightarrow S=7\)
Vậy \(S< \dfrac{1}{36}\)
\(S=\dfrac{1}{7^2}+\dfrac{2}{7^3}+\dfrac{3}{7^4}+...+\dfrac{69}{7^{70}}\)
\(\Rightarrow7S=\dfrac{1}{7}+\dfrac{2}{7^2}+\dfrac{3}{7^3}+...+\dfrac{69}{7^{69}}\)
\(\Rightarrow7S-S=\dfrac{1}{7}+\dfrac{2}{7^2}+\dfrac{3}{7^3}+...+\dfrac{69}{7^{69}}-\left(\dfrac{1}{7^2}+\dfrac{2}{7^3}+\dfrac{3}{7^4}+...+\dfrac{69}{7^{70}}\right)\)
\(\Rightarrow6S=\dfrac{1}{7}+\dfrac{1}{7^2}+\dfrac{1}{7^3}+...+\dfrac{1}{7^{69}}-\dfrac{69}{7^{70}}\) (*)
Đặt \(A=\dfrac{1}{7}+\dfrac{1}{7^2}+...+\dfrac{1}{7^{69}}\)
\(\Rightarrow7A=1+\dfrac{1}{7}+...+\dfrac{1}{7^{68}}\)
\(\Rightarrow7A-A=1+\dfrac{1}{7}+...+\dfrac{1}{7^{68}}-\left(\dfrac{1}{7}+\dfrac{1}{7^2}+...+\dfrac{1}{7^{69}}\right)\)
\(\Rightarrow6A=1-\dfrac{1}{7^{69}}\)
\(\Rightarrow A=\dfrac{7^{69}-1}{6.7^{69}}\)
(*) \(\Rightarrow6S=\dfrac{7^{69}-1}{6.7^{69}}-\dfrac{69}{7^{70}}=\dfrac{7^{70}-7-414}{6.7^{70}}=\dfrac{7^{70}-421}{6.7^{70}}\)
\(\Rightarrow S=\dfrac{7^{70}-421}{36.7^{70}}\)