Question

Resolver a equação x/3+x/9+x/27+x/81+...

Answer 1

Sendo uma PG infinita e de razão 0 > q > 1, fazemos o limite de soma:

$\mathsf{\lim S=\frac{a_1}{1-q}}\\\\\mathsf{\boxed{a_1=\frac{x}{3};q=\frac{1}{3}}}\\\\\mathsf{\lim S=\frac{\frac{x}{3}}{1-\frac{1}{3}}}\\\\\mathsf{\lim S=\frac{\frac{x}{3}}{\frac{3}{3}-\frac{1}{3}}}\\\\\mathsf{\lim S=\frac{\frac{x}{3}}{\frac{2}{3}}}\\\\\mathsf{\lim S=\frac{x}{3}\cdot\frac{3}{2}}\\\\\mathsf{\boxed{\lim S=\frac{x}{2}}}$