Giải thích các bước giải:
Ta có:
$S=\dfrac{1}{3}+\dfrac{4}{3^2}+\dfrac{7}{3^3}+...+\dfrac{2011}{3^{671}}$
$\to 3S=1+\dfrac{4}{3}+\dfrac{7}{3^2}+...+\dfrac{2011}{3^{670}}$
$\to 3S-S=1+\dfrac{4-1}{3}+\dfrac{7-4}{3^2}+...+\dfrac{2011-2008}{3^{670}}-\dfrac{2011}{3^{671}}$
$\to 2S=1+\dfrac{3}{3}+\dfrac{3}{3^2}+...+\dfrac{3}{3^{670}}-\dfrac{2011}{3^{671}}$
$\to 2S=1+1+\dfrac{1}{3}+...+\dfrac{1}{3^{669}}-\dfrac{2011}{3^{671}}$
$\to 2S=1+(1+\dfrac{1}{3}+...+\dfrac{1}{3^{669}})-\dfrac{2011}{3^{671}}$
$\to 2S=1+\dfrac{\dfrac{1}{3^{670}}-1}{\dfrac13-1}-\dfrac{2011}{3^{671}}$
$\to 2S=1+\dfrac{\dfrac{1}{3^{670}}-1}{-\dfrac23}-\dfrac{2011}{3^{671}}$
$\to 2S=\dfrac52-\dfrac{4031}{2\cdot 3^{671}}$
$\to S=\dfrac54-\dfrac{4031}{4\cdot 3^{671}}$
