Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 19

Apply the Leibniz rule for the $n$th derivative of a product. Only the first three terms survive, because all derivatives of $x^2$ of order greater than two vanish and every derivative of $e^x$ is $e^x$:At $n=100$ this gives

The epsilon-delta definition of continuity says that $f$ is continuous at $a$ ifHere $a$ belongs to the domain of $f$, and the implication is required for every $x$ in that domain. The number $\delta$ may depend on $\epsilon$ and $a$, but must work for all such $x$ simultaneously.

The bound $|\cos (1/x)|\le 1$ gives $|f(x)-f(0)|\le |x{|}^p$ when $x\ne 0$. If $p>0$, given any $\epsilon >0$, choose $\delta ={\epsilon }^{1/p}$. Then $0<|x|<\delta$ impliesthe inequality also holds at $x=0$. This proves continuity using the epsilon-delta definition of continuity.

For necessity, take the positive sequence $x_n=1/(2\pi n)\to 0$. Then $\cos (1/x_n)=1$, so $f(x_n)=(2\pi n{)}^{-p}$. For $p=0$ this equals $1$; for $p<0$ it tends to $+\infty$. In neither case does it tend to $f(0)=0$, so continuity fails. ConsequentlyThis is an instance of the general continuity and differentiability thresholds for a damped oscillation.

The difference quotient at zero isIf $p>1$, its absolute value is at most $|h{|}^{p-1}\to 0$, so $f$ is differentiable at zero with derivative $0$.

To exclude $p\le 1$, use $h_n=1/(2\pi n)$ and $k_n=1/(2\pi n+\pi /2)$. The difference quotient equals $(2\pi n{)}^{1-p}$ along the first sequence, and $0$ along the second. For $p=1$ the first value is $1$, while for $p<1$ it is unbounded. Hence a common finite limit is impossible. Therefore

For the positive terms $a_n=n^4/3^n$, the consecutive ratio isThe ratio test therefore proves that the series converges absolutely. More explicitly, for all sufficiently large $n$ the ratio is at most a fixed $q<1$, so the tail is bounded by a convergent geometric series.

Compare the positive terms with those of the harmonic series:By the limit comparison test, the given series has the same convergence behaviour as $\sum 1/n$, which diverges. Thus the series diverges to $+\infty$.