Question

Cho :S=\(1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+.......+\dfrac{1}{2013}-\dfrac{1}{2014}+\dfrac{1}{2015}\) và P=\(\dfrac{1}{1008}+\dfrac{1}{1009}+......+\dfrac{1}{2014}+\dfrac{1}{2015}\) Tính \(\left(S-P\right)^{2016}\)

Answer 1

Ta có:

\(S=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2013}-\dfrac{1}{2014}\)

\(=\left(1+\dfrac{1}{3}+...+\dfrac{1}{2015}\right)-\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2014}\right)\)

\(=\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2015}\right)-2\left(1+\dfrac{1}{2}+...+\dfrac{1}{2014}\right)\)

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

\(=\dfrac{1}{1008}+\dfrac{1}{1009}+\dfrac{1}{1010}+...+\dfrac{1}{2015}\)

Mà \(P=\dfrac{1}{1008}+\dfrac{1}{1009}+\dfrac{1}{1010}+...+\dfrac{1}{2015}\)

\(\Rightarrow S=P\Rightarrow S-P=0\)

\(\Rightarrow\left(S-P\right)^{2016}=0^{2016}=0\)

Vậy \(\left(S-P\right)^{2016}=0\)