证明:
∵x∈[0,π/2)
∴0<cosx≤1且单调递减.
于是对于任意n∈N,均有0<(cosx)^n≤1且单调递减.则必有
0<fn(x)=(cosx)^n+(cosx)^(n-1)+...+cosx≤n,且fn(x)在[0,π/2)单调递减.
由于fn(0)=n≥1,fn(π/2)=0,根据fn(x)的连续性及单调性,知必有唯一解x0∈[0,π/2)满足方程fn(x)=1.
证明:
∵x∈[0,π/2)
∴0<cosx≤1且单调递减.
于是对于任意n∈N,均有0<(cosx)^n≤1且单调递减.则必有
0<fn(x)=(cosx)^n+(cosx)^(n-1)+...+cosx≤n,且fn(x)在[0,π/2)单调递减.
由于fn(0)=n≥1,fn(π/2)=0,根据fn(x)的连续性及单调性,知必有唯一解x0∈[0,π/2)满足方程fn(x)=1.