问题
填空题
函数f(x)=lnx在x=n(n∈N*)处的切线斜率为an,则a1a2+a2a3+a3a4+…+a2010a2011=______.
答案
∵f′(x)=
,∴an=1 x
,1 n
∴a1a2+a2a3+a3a4+…+a2010a2011=
×1 1
+1 2
×1 2
+1 3
×1 3
+…+1 4
×1 2010 1 2011
=1-
+1 2
-1 2
+1 3
-1 3
+…+1 4
-1 2010 1 2011
=1-
=1 2011
.2010 2011
故答案:
.2010 2011