Question

[goo hye sun]

cho P = 1/5^2 + 2/5^3 + 3/5^4 + ... + 10/5^11 + 11/5^12 . CMR : P < 1/16

Answer 1

[Thanh Tùng DZ]

 P = 1/5^2 + 2/5^3 + 3/5^4 + ... + 10/5^11 + 11/5^12 . 

5P = \(\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+...+\frac{10}{5^{10}}+\frac{11}{5^{11}}\)

5P - P = ( \(\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+...+\frac{10}{5^{10}}+\frac{11}{5^{11}}\)) - ( 1/5^2 + 2/5^3 + 3/5^4 + ... + 10/5^11 + 11/5^12 .  )

4P = \(\left(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{11}}\right)-\frac{11}{5^{12}}\)

4P = \(\frac{1-\frac{1}{5^{11}}}{4}-\frac{11}{5^{12}}< \frac{1}{4}\)

\(P< \frac{1}{16}\)