= Right Riemann sum
{title2=$\sum_{j=1}^Nf(x_j)(x_j-x_{j-1})$}
A right <Riemann sum> on a partition $a=x_0<\cdots<x_N=b$ samples each subinterval at its right endpoint. For a continuous <function> its limit as the mesh tends to zero is the <integral>, by the <Riemann sum convergence theorem>. On a uniform partition of width $h$, comparison of endpoint weights in the <composite trapezoidal rule> gives $R_N-T_N=h(f(b)-f(a))/2$. Thus $N^{-5}\sum_{j=1}^N j^4$ is a right <Riemann sum> for $\int_0^1x^4\,dx$, and its limit is $1/5$.
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