grandes-ecoles 2011 QII.C

grandes-ecoles · France · centrale-maths1__psi Integration by Parts Prove an Integral Inequality or Bound

For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ Using yet another integration by parts, show that: $$\left| w_{k} - \frac{1}{12} \int_{k-1}^{k} \frac{\mathrm{~d}t}{t^{2}} \right| \leqslant \frac{1}{6} \int_{k-1}^{k} \frac{dt}{t^{3}}$$

For all integers $k \geqslant 2$, we denote:
$$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$
Using yet another integration by parts, show that:
$$\left| w_{k} - \frac{1}{12} \int_{k-1}^{k} \frac{\mathrm{~d}t}{t^{2}} \right| \leqslant \frac{1}{6} \int_{k-1}^{k} \frac{dt}{t^{3}}$$

Paper Questions

QI.A QI.B QI.C QI.D QII.A QII.B QII.C QII.D QIII.A QIII.B QIII.C QIII.D QIII.E QIV.A QIV.B QIV.C QIV.D QIV.E QV.A QV.B QV.C QV.D QVI.A QVI.B QVI.C