grandes-ecoles 2016 QIII.F

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series

If $\lambda > 1$, and $C_{n} = \sum_{k=0}^{n} \frac{(n\lambda)^{k}}{k!}$, determine an equivalent of $C_{n}$ when $n \rightarrow +\infty$.
Consider the integral $\frac{1}{n!} \int_{-\infty}^{0} (r - t)^{n} \mathrm{e}^{t} \mathrm{~d}t$ and choose the real number $r$ appropriately.

If $\lambda > 1$, and $C_{n} = \sum_{k=0}^{n} \frac{(n\lambda)^{k}}{k!}$, determine an equivalent of $C_{n}$ when $n \rightarrow +\infty$.

Consider the integral $\frac{1}{n!} \int_{-\infty}^{0} (r - t)^{n} \mathrm{e}^{t} \mathrm{~d}t$ and choose the real number $r$ appropriately.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QII.A QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIII.C.6 QIII.D.1 QIII.D.2 QIII.D.3 QIII.E.1 QIII.E.2 QIII.F