Respostas

  • Usuário do Brainly
2014-08-24T06:55:03-03:00
Inicialmente, observe que \dfrac{1}{k(k+1)}=\dfrac{1}{k}-\dfrac{1}{k+1}.

Logo,

\dfrac{1}{1\cdot2}=1-\dfrac{1}{2}, \dfrac{1}{2\cdot3}=\dfrac{1}{2}-\dfrac{1}{3}, \dots, \dfrac{1}{2~007\cdot2~008}=\dfrac{1}{2~007}-\dfrac{1}{2~008}.

Assim, temos

S=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dots+\dfrac{1}{2~006}-\dfrac{1}{2~007}+\dfrac{1}{2~007}-\dfrac{1}{2~008}

e, portanto, S=1-\dfrac{1}{2~008}=\dfrac{2~007}{2~008}.