Question

cho M=1-1/2+1/3-1/4+...+1/2015-1/2016+1/2017

N= 1/1009+1/1010+....+1/2016+1/2017

tính (M-N)^2017

Answer 1

\(M=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2015}-\dfrac{1}{2016}+\dfrac{1}{2017}\)

\(M=\left(1+\dfrac{1}{3}+...+\dfrac{1}{2015}+\dfrac{1}{2017}\right)-\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2016}\right)\)\(M=1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}...+\dfrac{1}{2015}+\dfrac{1}{2016}+\dfrac{1}{2017}-2\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2016}\right)\)\(M=1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2015}+\dfrac{1}{2016}+\dfrac{1}{2017}-\left(1+\dfrac{1}{2}+\dfrac{1}{3}+.....+\dfrac{1}{2008}\right)\)

\(M=\dfrac{1}{2009}+\dfrac{1}{2010}+...+\dfrac{1}{2016}+\dfrac{1}{2017}=N\)

Vậy \(\left(M-N\right)^{2017}=0\)