Evaluate the following:

Question

  lim     1      1       2       3             n

           — [ (—)⁹+(—)⁹+(—)⁹+...+(—)⁹ ]

n->∞   n      n       n       n             n

Answer 1

The given limit can be written as:

 

   lim_(n→∞)  ∑_{(from i=1 to i=n)}[f(i/n)(1/n)], where f(x) = x⁹.

 

= ∫_{(from 0 to 1)} x⁹ dx = (1/10)[x¹⁰]_{(from 0 to 1)} = 1/10