Past exam of the approximation theory and finite element methods course of the University of Oxford / 2010 / A / 2 / d / Solution

For $n\ge 1$, ${\varphi }_n$ is orthogonal to every polynomial in $P_{n-1}$, so $(u,{\varphi }_n)=(u-p_{n-1}^{\ast },{\varphi }_n)$. The Cauchy-Schwarz inequality givesUsing the tail identity from part (b) and shifting the index,For the stated power law and $n>1$, the decreasing-function integral test givesConsequentlyThe assumption $a>1$ is sufficient; this summation estimate itself only needs $a>1/2$.