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GATE 2025 MA – Question 56

Real Analysis · Sequences and Series of Functions · 2 marks · Multiple select

For x∈(0,π), u_n(x)=sin(nx)/√n. Which statements about the series hold?

  1. Σu_n converges uniformly on (0,π).
  2. Σu_n′ converges uniformly on (0,π).
  3. Σu_n converges pointwise but not uniformly on (0,π).
  4. Σu_n converges uniformly on every compact subset of (0,π).

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Correct answer: (C) Σu_n converges pointwise but not uniformly on (0,π).; (D) Σu_n converges uniformly on every compact subset of (0,π).

Explanation

Dirichlet’s test gives pointwise convergence; sine partial sums are uniformly bounded away from x=0, so it also gives uniform convergence on compact subsets, proving D. For x=1/(2N) and N≤n≤2N, sin(nx)≥sin(1/2), so the tail sum is at least (N+1)sin(1/2)/√(2N), violating uniform Cauchy convergence. Thus C holds. Derivative terms √n cos(nx) do not even tend uniformly to zero, refuting B. **C,D**.

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