一道数学证明题
求证：(1/2^10)+(1/(2^10+1))+(1/(2^10+2))+...+(1/(2^11+1))<1
参考答案:(1/2^10)+(1/(2^10+1))+(1/(2^10+2))+...+(1/(2^11+1))=1/1024+1/1025+1/1026+........1/2049
设;0<t<=512
1/(1024+t)+1/(2049-t)=3073/{(1024+t)*(2049-t)}
=3073/{1024*2049+1025*t-t*t}
=3073/{1024*2079+513*t+t*(512-t)}<3073/(1024*2049)
=1/1024+1/2049
所以
(1/2^10)+(1/(2^10+1))+(1/(2^10+2))+...+(1/(2^11+1))<513*(1/1024+1/2049)=513*3073/(1024*2049)<1