Question

Question #0debb

Answer 1

Let us look at the definition of the Riemann definite integral:
`int_a^bf(x)\ dx=(b-a)lim_(N->oo)1/Nsum_(i=1)^Nf(a+(i(b-a))/N)`

With this, we shall manipulate the sum in question into a form that can be translated to the definite integral.

`\ \ \ \ \ \ lim_(n->oo)sum_(i=1)^N(N+i)/N^2`
`=lim_(n->oo)1/Nsum_(i=1)^N(N+i)/N`
`=lim_(n->oo)1/Nsum_(i=1)^N(1+i/N)`
`=(1-0)lim_(n->oo)1/Nsum_(i=1)^N(1+(0+(i(1-0))/N))`
`=int_0^1(1+x)\ dx`
`=[(1+x)^2/2]_0^1`
`=3/2`