← 2019 Paper 2

UPSC 2019 Maths Optional Paper 2 Q3a — Step-by-Step Solution

15 marks · Section A

Pointwise vs. Uniform Convergence of Sequences of Functions · Real Analysis · Read the full method →

Question

Discuss the uniform convergence of

Technique

Pointwise limit, then sup-norm (Weierstrass $M_n$) test: locate the maximizer $x=1/n$ by calculus, evaluate the peak height $\tfrac12$.

Solution

Step 1 — Pointwise limit

Fix $x\in\mathbb R$.

• If $x=0$: $f_n(0)=0$ for all $n$, so $f_n(0)\to0$.
• If $x\ne0$: $f_n(x)=\dfrac{nx}{1+n^2x^2}=\dfrac{x/n}{1/n^2+x^2}\to\dfrac{0}{x^2}=0$ as $n\to\infty$.

So $f_n\to f\equiv0$ pointwise on $\mathbb R$.

Step 2 — Sup-norm test for uniform convergence on $\mathbb R$

Uniform convergence to $0$ requires $M_n:=\sup_{x\in\mathbb R}|f_n(x)-0|\to0$. Maximize $|f_n|$. By symmetry consider $x>0$; set $g(x)=\dfrac{nx}{1+n^2x^2}$ and solve $g'(x)=0$:

At $x=1/n$,

Hence

Step 3 — Conclusion on $\mathbb R$

$M_n=\tfrac12\not\to0$. Therefore the convergence is not uniform on $\mathbb R$.

Answer