So sánh P với 1/2 biết P=3/(1!+2!+3!) + 4/(2!+3!+4!) + …+ 2017/(2015!+2016!+2017!) (với n!=1.2.3…n)

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1. Giải thích các bước giải:

   Ta có:

   $\dfrac{k+2}{k!+\left(k+1\right)!+\left(k+2\right)!}$

   $=\dfrac{k+2}{k!\left(1+\left(k+1\right)+\left(k+1\right)\left(k+2\right)\right)}$

   $=\dfrac{k+2}{k!\left(\left(k+2\right)+\left(k+1\right)\left(k+2\right)\right)}$

   $=\dfrac{k+2}{k!\left(k+2\right)\left(1+\left(k+1\right)\right)}$

   $=\dfrac{k+2}{k!\left(k+2\right)\left(k+2\right)}$

   $=\dfrac{1}{k!\left(k+2\right)}$

   $=\dfrac{k+1}{k!\left(k+2\right)\left(k+1\right)}$

   $=\dfrac{k+1}{\left(k+2\right)!}$

   $=\dfrac{\left(k+2\right)-1}{\left(k+2\right)!}$

   $=\dfrac{1}{\left(k+1\right)!}-\dfrac{1}{\left(k+2\right)!}$

   Áp dụng  công thức trên

   $\to P=\dfrac1{2!}-\dfrac1{3!}+\dfrac1{3!}-\dfrac1{4!}+….+\dfrac1{2016!}-\dfrac{1}{2017!}$

   $\to P=\dfrac1{2!}-\dfrac{1}{2017!}$

   $\to P<\dfrac1{2!}$

   $\to P<\dfrac12$

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