Year: 2021
Paper: 2
Question Number: 8
Course: LFM Pure
Section: Integration
No solution available for this problem.
Candidates were generally well prepared for many of the questions on this paper, with the questions requiring more standard operations seeing the greatest levels of success. Candidates need to ensure that solutions to the questions are supported by sufficient evidence of the mathematical steps, for example when proving a given result or deducing the properties of graphs that are to be sketched. In a significant number of steps there were marks lost through simple errors such as mistakes in arithmetic or confusion of sine and cosine functions, so it is important for candidates to maintain accuracy in their solutions to these questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Show that, for $n = 2, 3, 4, \ldots$,
\[
\frac{d^2}{dt^2}\bigl[t^n(1-t)^n\bigr] = n\,t^{n-2}(1-t)^{n-2}\bigl[(n-1) - 2(2n-1)t(1-t)\bigr].
\]
\item The sequence $T_0, T_1, \ldots$ is defined by
\[
T_n = \int_0^1 \frac{t^n(1-t)^n}{n!}\,e^t\,dt.
\]
Show that, for $n \geqslant 2$,
\[
T_n = T_{n-2} - 2(2n-1)T_{n-1}.
\]
\item Evaluate $T_0$ and $T_1$ and deduce that, for $n \geqslant 0$, $T_n$ can be written in the form
\[
T_n = a_n + b_n e,
\]
where $a_n$ and $b_n$ are integers (which you should not attempt to evaluate).
\item Show that $0 < T_n < \dfrac{e}{n!}$ for $n \geqslant 0$. Given that $b_n$ is non-zero for all~$n$, deduce that $\dfrac{-a_n}{b_n}$ tends to $e$ as $n$ tends to infinity.
\end{questionparts}
Many candidates were able to complete the differentiation correctly in terms of n for part (i) of this question although in some cases the result that was given was jumped to with insufficient justification following the completion of the differentiation. Similarly, in part (ii) many candidates chose appropriate methods to reach the required relation, but did not provide sufficient working to show that the appropriate manipulations had been carried out correctly. Candidates need to ensure that solutions to questions where the answer is given provide sufficiently detailed explanation of the steps that are taken. In part (iii) many candidates were able to evaluate the necessary base cases for the proof by induction and provided some justification for the inductive step, although in some cases it was not sufficiently clear that the values of a_n and b_n would be integers. Many candidates were then able to demonstrate some understanding of the necessary steps for the final part, but in some cases insufficient detail was present to secure full marks.