Question

2 c)
20 pct + Coroana :)

Answer 1

$\displaystyle Conform~punctului~b)~avem~sin^3 \frac{x}{3^k}= \frac{1}{4} \left(3sin \frac{x}{3^k}-sin \frac{x}{3^{k-1}} \right). \\ \\ S= \frac{1}{4} \sum\limits^n_{k=1} \left(3^k sin \frac{x}{3^k}-3^{k-1} sin \frac{x}{3^{k-1}} \right)= \\ \\ = \frac{1}{4} \Big( 3sin \frac{x}{3}-sinx+3^2 sin \frac{x}{3^2}-3sin \frac{x}{3}+ 3^3sin\frac{x}{3^3}-3^2sin \frac{x}{3^2}+... \\ \\ +3^nsin \frac{x}{3^n}-3^{n-1}sin \frac{x}{3^{n-1}} \Big)= \\ \\ = \frac{1}{4} \left( 3^nsin \frac{x}{3^n}-sinx \right)$