回復 1# Christina 的帖子
第 8 題
\(f(x)=a_1sinx+a_2sin2x+\ldots+a_n sinnx\),\(a_i\in R,n\in N\)且\(|\;f(x)|\;\le |\;sinx|\;\),\(\forall x\in R\)證明:\(|\;a_1+2a_2+\ldots+na_n|\;\le 1\)
[解答]
\(\left| {{a}_{1}}+2{{a}_{2}}+\cdots +n{{a}_{n}} \right|=\left| f'\left( 0 \right) \right|=\underset{x\to 0}{\mathop{\lim }}\,\left| \frac{f\left( x \right)-f\left( 0 \right)}{x-0} \right|=\underset{x\to 0}{\mathop{\lim }}\,\left| \frac{f\left( x \right)}{x} \right|\le \underset{x\to 0}{\mathop{\lim }}\,\left| \frac{\sin x}{x} \right|=1\)